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Khachatour Koshtoyants (Armenian: Խաչատուր Սեդրակի Կոշտոյանց; Russian: Хачатур Седракович Коштоянц; September 26, 1900 – April 2, 1961) was a Soviet physiologist, Corresponding Member of the Academy of Sciences of the Soviet Union (since 1939), Member of the Armenian National Academy of Sciences (since 1943), Professor at the Lomonosov Moscow State University (since 1930), Doktor Nauk in Biological Sciences (1935). He was a Laureate of the 1947 Stalin Prize. == Life == He was born in Armenia. He graduated from the Lomonosov Moscow State University in 1926. From 1929 he worked in his alma mater. In 1935 he received the title of Professor. From 1943 Koshtoyants headed the Department of Physiology of Animals at the Lomonosov Moscow State University. Also from 1936 he worked at the Severtsov Institute of Ecology and Evolution of the Academy of Sciences of the Soviet Union. From 1946 to 1953 he was Director of the Vavilov Institute for the History of Science and Technology. He died in Moscow in 1961. == References ==
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The cyclol hypothesis is the now discredited first structural model of a folded, globular protein, formulated in the 1930s. It was based on the cyclol reaction of peptide bonds proposed by physicist Charles Frank in 1936, in which two peptide groups are chemically crosslinked. These crosslinks are covalent analogs of the non-covalent hydrogen bonds between peptide groups and have been observed in rare cases, such as the ergopeptides. Based on this reaction, mathematician Dorothy Wrinch hypothesized in a series of five papers in the late 1930s a structural model of globular proteins. She postulated that, under some conditions, amino acids will spontaneously make the maximum possible number of cyclol crosslinks, resulting in cyclol molecules and cyclol fabrics. She further proposed that globular proteins have a tertiary structure corresponding to Platonic solids and semiregular polyhedra formed of cyclol fabrics with no free edges. In contrast to the cyclol reaction itself, these hypothetical molecules, fabrics and polyhedra have not been observed experimentally. The model has several consequences that render it energetically implausible, such as steric clashes between the protein sidechains. In response to such criticisms J. D. Bernal proposed that hydrophobic interactions are chiefly responsible for protein folding, which was indeed borne out. == Historical context == By the mid-1930s, analytical ultracentrifugation studies by Theodor Svedberg had shown that proteins had a well-defined chemical structure, and were not aggregations of small molecules. The same studies appeared to show that the molecular weight of proteins fell into a few well-defined classes related by integers, such as Mw = 2p3q Da, where p and q are nonnegative integers. However, it was difficult to determine the exact molecular weight and number of amino acids in a protein. Svedberg had also shown that a change in solution conditions could cause a protein to disassemble into small
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subunits, now known as a change in quaternary structure. The chemical structure of proteins was still under debate at that time. The most accepted (and ultimately correct) hypothesis was that proteins are linear polypeptides, i.e., unbranched polymers of amino acids linked by peptide bonds. However, a typical protein is remarkably long—hundreds of amino-acid residues—and several distinguished scientists were unsure whether such long, linear macromolecules could be stable in solution. Further doubts about the polypeptide nature of proteins arose because some enzymes were observed to cleave proteins but not peptides, whereas other enzymes cleave peptides but not folded proteins. Attempts to synthesize proteins in the test tube were unsuccessful, mainly due to the chirality of amino acids; naturally occurring proteins are composed of only left-handed amino acids. Hence, alternative chemical models of proteins were considered, such as the diketopiperazine hypothesis of Emil Abderhalden. However, no alternative model had yet explained why proteins yield only amino acids and peptides upon hydrolysis and proteolysis. As clarified by Linderstrøm-Lang, these proteolysis data showed that denatured proteins were polypeptides, but no data had yet been obtained about the structure of folded proteins; thus, denaturation could involve a chemical change that converted folded proteins into polypeptides. The process of protein denaturation (as distinguished from coagulation) had been discovered in 1910 by Harriette Chick and Charles Martin, but its nature was still mysterious. Tim Anson and Alfred Mirsky had shown that denaturation was a reversible, two-state process that results in many chemical groups becoming available for chemical reactions, including cleavage by enzymes. In 1929, Hsien Wu hypothesized correctly that denaturation corresponded to protein unfolding, a purely conformational change that resulted in the exposure of amino-acid side chains to the solvent. Wu's hypothesis was also advanced independently in 1936 by Mirsky and Linus Pauling. Nevertheless, protein scientists could
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not exclude the possibility that denaturation corresponded to a chemical change in the protein structure, a hypothesis that was considered a (distant) possibility until the 1950s. X-ray crystallography had just begun as a discipline in 1911, and had advanced relatively rapidly from simple salt crystals to crystals of complex molecules such as cholesterol. However, even the smallest proteins have over 1000 atoms, which makes determining their structure far more complex. In 1934, Dorothy Crowfoot Hodgkin had taken crystallographic data on the structure of the small protein, insulin, although the structure of that and other proteins were not solved until the late 1960s. However, pioneering X-ray fiber diffraction data had been collected in the early 1930s for many natural fibrous proteins such as wool and hair by William Astbury, who suggested that "globular proteins in general might be folded from elements essentially like the elements of fibrous proteins." Since protein structure was so poorly understood in the 1930s, the physical interactions responsible for stabilizing that structure were likewise unknown. Astbury hypothesized that the structure of fibrous proteins was stabilized by hydrogen bonds in β-sheets. The idea that globular proteins are also stabilized by hydrogen bonds was proposed by Dorothy Jordan Lloyd in 1932, and championed later by Alfred Mirsky and Linus Pauling. At a 1933 lecture by Astbury to the Oxford Junior Scientific Society, physicist Charles Frank suggested that the fibrous protein α-keratin might be stabilized by an alternative mechanism, namely, covalent crosslinking of the peptide bonds by the cyclol reaction above. The cyclol crosslink draws the two peptide groups close together; the N and C atoms are separated by ~1.5 Å, whereas they are separated by ~3 Å in a typical hydrogen bond. The idea intrigued J. D. Bernal, who suggested it to the mathematician Dorothy Wrinch as possibly useful
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in understanding protein structure. == Basic theory == Wrinch developed this suggestion into a full-fledged model of protein structure. The basic cyclol model was laid out in her first paper (1936). She noted the possibility that polypeptides might cyclize to form closed rings (true) and that these rings might form internal crosslinks through the cyclol reaction (also true, although rare). Assuming that the cyclol form of the peptide bond could be more stable than the amide form, Wrinch concluded that certain cyclic peptides would naturally make the maximal number of cyclol bonds (such as cyclol 6, Figure 2). Such cyclol molecules would have hexagonal symmetry, if the chemical bonds were taken as having the same length, roughly 1.5 Å; for comparison, the N-C and C-C bonds have the lengths 1.42 Å and 1.54 Å, respectively. These rings can be extended indefinitely to form a cyclol fabric (Figure 3). Such fabrics exhibit a long-range, quasi-crystalline order that Wrinch felt was likely in proteins, since they must pack hundreds of residues densely. Another interesting feature of such molecules and fabrics is that their amino-acid side chains point axially upwards from only one face; the opposite face has no side chains. Thus, one face is completely independent of the primary sequence of the peptide, which Wrinch conjectured might account for sequence-independent properties of proteins. In her initial article, Wrinch stated clearly that the cyclol model was merely a working hypothesis, a potentially valid model of proteins that would have to be checked. Her goals in this article and its successors were to propose a well-defined testable model, to work out the consequences of its assumptions and to make predictions that could be tested experimentally. In these goals, she succeeded; however, within a few years, experiments and further modeling showed that the cyclol hypothesis
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was untenable as a model for globular proteins. == Stabilizing energies == In two tandem Letters to the Editor (1936), Wrinch and Frank addressed the question of whether the cyclol form of the peptide group was indeed more stable than the amide form. A relatively simple calculation showed that the cyclol form is significantly less stable than the amide form. Therefore, the cyclol model would have to be abandoned unless a compensating source of energy could be identified. Initially, Frank proposed that the cyclol form might be stabilized by better interactions with the surrounding solvent; later, Wrinch and Irving Langmuir hypothesized that hydrophobic association of nonpolar sidechains provides stabilizing energy to overcome the energetic cost of the cyclol reactions. The lability of the cyclol bond was seen as an advantage of the model, since it provided a natural explanation for the properties of denaturation; reversion of cyclol bonds to their more stable amide form would open up the structure and allows those bonds to be attacked by proteases, consistent with experiment. Early studies showed that proteins denatured by pressure are often in a different state than the same proteins denatured by high temperature, which was interpreted as possibly supporting the cyclol model of denaturation. The Langmuir-Wrinch hypothesis of hydrophobic stabilization shared in the downfall of the cyclol model, owing mainly to the influence of Linus Pauling, who favored the hypothesis that protein structure was stabilized by hydrogen bonds. Another twenty years had to pass before hydrophobic interactions were recognized as the chief driving force in protein folding. == Steric complementarity == In her third paper on cyclols (1936), Wrinch noted that many "physiologically active" substances such as steroids are composed of fused hexagonal rings of carbon atoms and, thus, might be sterically complementary to the face of cyclol molecules without
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the amino-acid side chains. Wrinch proposed that steric complementarity was one of chief factors in determining whether a small molecule would bind to a protein. Wrinch speculated that proteins are responsible for the synthesis of all biological molecules. Noting that cells digest their proteins only under extreme starvation conditions, Wrinch further speculated that life could not exist without proteins. == Hybrid models == From the beginning, the cyclol reaction was considered as a covalent analog of the hydrogen bond. Therefore, it was natural to consider hybrid models with both types of bonds. This was the subject of Wrinch's fourth paper on the cyclol model (1936), written together with Dorothy Jordan Lloyd, who first proposed that globular proteins are stabilized by hydrogen bonds. A follow-up paper was written in 1937 that referenced other researchers on hydrogen bonding in proteins, such as Maurice Loyal Huggins and Linus Pauling. Wrinch also wrote a paper with William Astbury, noting the possibility of a keto-enol isomerization of the >CαHα and an amide carbonyl group >C=O, producing a crosslink >Cα-C(OHα)< and again converting the oxygen to a hydroxyl group. Such reactions could yield five-membered rings, whereas the classic cyclol hypothesis produces six-membered rings. This keto-enol crosslink hypothesis was not developed much further. == Space-enclosing fabrics == In her fifth paper on cyclols (1937), Wrinch identified the conditions under which two planar cyclol fabrics could be joined to make an angle between their planes while respecting the chemical bond angles. She identified a mathematical simplification, in which the non-planar six-membered rings of atoms can be represented by planar "median hexagon"s made from the midpoints of the chemical bonds. This "median hexagon" representation made it easy to see that the cyclol fabric planes can be joined correctly if the dihedral angle between the planes equals the tetrahedral bond
|
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angle δ = arccos(-1/3) ≈ 109.47°. A large variety of closed polyhedra meeting this criterion can be constructed, of which the simplest are the truncated tetrahedron, the truncated octahedron, and the octahedron, which are Platonic solids or semiregular polyhedra. Considering the first series of "closed cyclols" (those modeled on the truncated tetrahedron), Wrinch showed that their number of amino acids increased quadratically as 72n2, where n is the index of the closed cyclol Cn. Thus, the C1 cyclol has 72 residues, the C2 cyclol has 288 residues, etc. Preliminary experimental support for this prediction came from Max Bergmann and Carl Niemann, whose amino-acid analyses suggested that proteins were composed of integer multiples of 288 amino-acid residues (n=2). More generally, the cyclol model of globular proteins accounted for the early analytical ultracentrifugation results of Theodor Svedberg, which suggested that the molecular weights of proteins fell into a few classes related by integers. The cyclol model was consistent with the general properties then attributed to folded proteins. (1) Centrifugation studies had shown that folded proteins were significantly denser than water (~1.4 g/mL) and, thus, tightly packed; Wrinch assumed that dense packing should imply regular packing. (2) Despite their large size, some proteins crystallize readily into symmetric crystals, consistent with the idea of symmetric faces that match up upon association. (3) Proteins bind metal ions; since metal-binding sites must have specific bond geometries (e.g., octahedral), it was plausible to assume that the entire protein also had similarly crystalline geometry. (4) As described above, the cyclol model provided a simple chemical explanation of denaturation and the difficulty of cleaving folded proteins with proteases. (5) Proteins were assumed to be responsible for the synthesis of all biological molecules, including other proteins. Wrinch noted that a fixed, uniform structure would be useful for proteins in templating
|
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their own synthesis, analogous to the Watson-Francis Crick concept of DNA templating its own replication. Given that many biological molecules such as sugars and sterols have a hexagonal structure, it was plausible to assume that their synthesizing proteins likewise had a hexagonal structure. Wrinch summarized her model and the supporting molecular-weight experimental data in three review articles. == Predicted protein structures == Having proposed a model of globular proteins, Wrinch investigated whether it was consistent with the available structural data. She hypothesized that bovine tuberculin protein (523) was a C1 closed cyclol consisting of 72 residues and that the digestive enzyme pepsin was a C2 closed cyclol of 288 residues. These residue-number predictions were difficult to verify, since the methods then available to measure the mass of proteins were inaccurate, such as analytical ultracentrifugation and chemical methods. Wrinch also predicted that insulin was a C2 closed cyclol consisting of 288 residues. Limited X-ray crystallographic data were available for insulin which Wrinch interpreted as "confirming" her model. However, this interpretation drew rather severe criticism for being premature. Careful studies of the Patterson diagrams of insulin taken by Dorothy Crowfoot Hodgkin showed that they were roughly consistent with the cyclol model; however, the agreement was not good enough to claim that the cyclol model was confirmed. == Implausibility of the model == The cyclol fabric was shown to be implausible for several reasons. Hans Neurath and Henry Bull showed that the dense packing of side chains in the cyclol fabric was inconsistent with the experimental density observed in protein films. Maurice Huggins calculated that several non-bonded atoms of the cyclol fabric would approach more closely than allowed by their van der Waals radii; for example, the inner Hα and Cα atoms of the lacunae would be separated by only 1.68 Å (Figure
|
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5). Haurowitz showed chemically that the outside of proteins could not have a large number of hydroxyl groups, a key prediction of the cyclol model, whereas Meyer and Hohenemser showed that cyclol condensations of amino acids did not exist even in minute quantities as a transition state. More general chemical arguments against the cyclol model were given by Bergmann and Niemann and by Neuberger. Infrared spectroscopic data showed that the number of carbonyl groups in a protein did not change upon hydrolysis, and that intact, folded proteins have a full complement of amide carbonyl groups; both observations contradict the cyclol hypothesis that such carbonyls are converted to hydroxyl groups in folded proteins. Finally, proteins were known to contain proline in significant quantities (typically 5%); since proline lacks the amide hydrogen and its nitrogen already forms three covalent bonds, proline seems incapable of the cyclol reaction and of being incorporated into a cyclol fabric. An encyclopedic summary of the chemical and structural evidence against the cyclol model was given by Pauling and Niemann. Moreover, a supporting piece of evidence—the result that all proteins contain an integer multiple of 288 amino-acid residues—was likewise shown to be incorrect in 1939. Wrinch replied to the steric-clash, free-energy, chemical and residue-number criticisms of the cyclol model. On steric clashes, she noted that small deformations of the bond angles and bond lengths would allow these steric clashes to be relieved, or at least reduced to a reasonable level. She noted that distances between non-bonded groups within a single molecule can be shorter than expected from their van der Waals radii, e.g., the 2.93 Å distance between methyl groups in hexamethylbenzene. Regarding the free-energy penalty for the cyclol reaction, Wrinch disagreed with Pauling's calculations and stated that too little was known of intramolecular energies to rule out
|
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the cyclol model on that basis alone. In reply to the chemical criticisms, Wrinch suggested that the model compounds and simple bimolecular reactions studied need not pertain to the cyclol model, and that steric hindrance may have prevented the surface hydroxyl groups from reacting. On the residue-number criticism, Wrinch extended her model to allow for other numbers of residues. In particular, she produced a "minimal" closed cyclol of only 48 residues, and, on that (incorrect) basis, may have been the first to suggest that the insulin monomer had a molecular weight of roughly 6000 Da. Therefore, she maintained that the cyclol model of globular proteins was still potentially viable and even proposed the cyclol fabric as a component of the cytoskeleton. However, most protein scientists ceased to believe in it and Wrinch turned her scientific attention to mathematical problems in X-ray crystallography, to which she contributed significantly. One exception was physicist Gladys Anslow, Wrinch's colleague at Smith College, who studied the ultraviolet absorption spectra of proteins and peptides in the 1940s and allowed for the possibility of cyclols in interpreting her results. As the sequence of insulin began to be determined by Frederick Sanger, Anslow published a three-dimensional cyclol model with sidechains, based on the backbone of Wrinch's 1948 "minimal cyclol" model. == Partial redemption == The downfall of the overall cyclol model generally led to a rejection of its elements; one notable exception was J. D. Bernal's short-lived acceptance of the Langmuir-Wrinch hypothesis that protein folding is driven by hydrophobic association. Nevertheless, cyclol bonds were identified in small, naturally occurring cyclic peptides in the 1950s. Clarification of the modern terminology is appropriate. The classic cyclol reaction is the addition of the NH amine of a peptide group to the C=O carbonyl group of another; the resulting compound is now
|
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called an azacyclol. By analogy, an oxacyclol is formed when an OH hydroxyl group is added to a peptidyl carbonyl group. Likewise, a thiacyclol is formed by adding an SH thiol moiety to a peptidyl carbonyl group. The oxacyclol alkaloid ergotamine from the fungus Claviceps purpurea was the first identified cyclol. The cyclic depsipeptide serratamolide is also formed by an oxacyclol reaction. Chemically analogous cyclic thiacyclols have also been obtained. Classic azacyclols have been observed in small molecules and tripeptides. Peptides are naturally produced from the reversion of azacylols, a key prediction of the cyclol model. Hundreds of cyclol molecules have now been identified, despite Linus Pauling's calculation that such molecules should not exist because of their unfavorably high energy. After a long hiatus during which she worked mainly on the mathematics of X-ray crystallography, Wrinch responded to these discoveries with renewed enthusiasm for the cyclol model and its relevance in biochemistry. She also published two books describing the cyclol theory and small peptides in general. == References == == Further reading ==
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Project SHAD, an acronym for Shipboard Hazard and Defense, was part of a larger effort called Project 112, which was conducted during the 1960s. Project SHAD encompassed tests designed to identify U.S. warships' vulnerabilities to attacks with chemical agents or biological warfare agents and to develop procedures to respond to such attacks while maintaining a war-fighting capability. == History == Project SHAD was part of a larger effort by the Department of Defense called Project 112. Project 112 was a chemical and biological weapons research, development, and testing project conducted by the United States Department of Defense and CIA handled by the Deseret Test Center and United States Army Chemical Materials Agency from 1962 to 1973. The project started under John F. Kennedy's administration, and was authorized by his Secretary of Defense Robert McNamara, as part of a total review of the US military. The name of the project refers to its number in the 150 review process. == Mission == The Shipboard Hazard and Defense Project (SHAD) was a series of tests conducted by the U.S. Department of Defense during the 1960s to determine how well service members aboard military ships could detect and respond to chemical and biological attacks. Dee Dodson Morris of the Army Chemical Corps who coordinated the ongoing investigation, says, "The SHAD tests were intended to show how vulnerable Navy ships were to chemical or biological warfare agents. The objective was to learn how chemical or biological warfare agents would disperse throughout a ship, and to use that information to develop procedures to protect crew members and decontaminate ships." DoD investigators note that over a hundred tests were planned but the lack of test results may indicate that many tests were never actually executed. 134 tests were planned initially, but reportedly, only 46 tests were
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actually completed. == Declassification == Public Law 107–314 required the identification and release of not only Project 112 information to the United States Veterans Administration, but also that of any other projects or tests where a veteran might have been exposed to a chemical or biological warfare agent, and directed the Secretary of Defense to work with veterans and veterans service organizations to identify the other projects or tests conducted by the Department of Defense that may have exposed members of the Armed Forces to chemical or biological agents. In 2000, the Department of Defense began the process of declassifying records about the project. According to the U.S. Department of Veteran Affairs, approximately 6,000 U.S. Service members were believed to be involved in conducting the tests. In 2002, the Department of Defense began publishing a list of fact sheets for each of the tests. == Identification == Although many of the roughly 5,500 veterans who took part were aware of the tests, some were involved without their knowledge. Certain issues surrounding the test program were not resolved by the passage of the law and the Department of Defense was accused of continuing to withhold documents on Cold War chemical and biological weapons tests that used unsuspecting veterans as "human samplers" after reporting to Congress it had released all medically relevant information. A Government Accountability Office May 2004 report, Chemical and Biological Defense: DOD Needs to Continue to Collect and Provide Information on Tests and Potentially Exposed Personnel indicates that almost all participants who were identified from Project 112 — 94 percent — were from ship-based tests of Project SHAD that comprised only about one-third of the total number of tests conducted. == Medical studies == Jack Alderson, a retired Navy officer who commanded the Army tugboats, told CBS News that
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he believes the Pentagon used him and other service members to test weapons, and that those tests included agents, vaccines, and decontamination products which have led to serious medical problems, including cancer. Secrecy agreements can now be ignored by veterans in order to pursue healthcare concerns within the Department of Veterans Affairs. The V.A. has offered screening programs for veterans who believe they were involved in DoD sponsored tests during their service. The Institute of Medicine of the National Academies has commissioned studies of Project SHAD participants. The first, Long-Term Health Effects of Participation in Project SHAD (Shipboard Hazard and Defense), was released in 2007, and found "no clear evidence that specific long-term health effects are associated with participation in Project SHAD." The second, Shipboard Hazard and Defense II (SHAD II), by the IoM's Medical Follow-up Agency (MFUA), began in 2012, and, as of April 2014, was ongoing. == Locations of SHAD tests == from SHAD Fact Sheets Pacific Ocean Pacific Ocean off San Diego, CA Pacific Ocean, from San Diego, CA to Balboa, Panama Ft. Sherman, Panama Canal Zone Eniwetok Atoll, Marshall Islands Baker Island Island of Hawaii Pacific Ocean near the Island of Oahu, Hawaii Pacific Ocean, west of Oahu, Hawaii Pacific Ocean southwest of Oahu, HI Oahu, HI and surrounding waters Pacific Ocean, off Oahu, HI & surrounding water & airspace Pacific Ocean near Johnston Island Eglin Air Force Base, Florida Atlantic Ocean off Newfoundland, Canada == Ships and air units involved == === Ships === USS Berkeley (DDG-15) USS Carbonero (SS-337) USS Carpenter (DD-825) USS Fechteler (DD-870) USS Fort Snelling (LSD-30) USS George Eastman (YAG-39) USS Granville S. Hall (YAG-40) USS Herbert J. Thomas (DD-833) USS Hoel (DDG-13) USS Moctobi (ATF-105) USS Navarro (APA-215) USS Okanogan (APA-220) USS Power (DD-839) USNS Samuel Phillips Lee (T-AGS 31)
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USNS Silas Bent (T-AGS-26) USS Tioga County (LST-1158) USS Tiru (SS-416) USS Wexford County (LST-1168) US Navy Lighter Barge (YFN-811) US Army Large Tug LT-2080 US Army Large Tug LT-2081 US Army Large Tug LT-2085 US Army Large Tug LT-2086 US Army Large Tug LT-2087 US Army Large Tug LT-2088 === Air units === Meteorological Research Service (contractor) C-47 US Navy Airborne Early Warning Barrier Squadron Pacific (AEWBarRonpac) Lockheed WV-2 / EC-121 Warning Star US Navy Patrol Squadron Four (VP-4) (PATRON Four), Fleet Air Wing Four (Lockheed P-2 Neptune) US Navy Patrol Squadron Six (VP-6) (PATRON SIX), Fleet Air Wing Two (Lockheed P-2 Neptune) US Navy VU-1 Utility Squadron One, re-designated VC-1 Composite Squadron One on July 1, 1965. (Douglas A-4 Skyhawk) USMC Marine Aircraft Group 13, 1st Marine Aircraft Wing, 1st Marine Expeditionary Brigade. McDonnell Douglas F-4 Phantom II, A-4 (Skyhawk), Sikorsky H-34 Seahorse USMC Marine Medium Helicopter Squadron 161 (HMM 161), Marine Aircraft Group 16 (MAG-16), 3rd Marine Aircraft Wing (3rd MAW) Sikorsky H-34 Seahorse US Air Force 4533rd Tactical Test Squadron, US Air Force 33rd Tactical Fighter Wing (F-4E Phantom II) == See also == Human experimentation in the United States Operation Dew Operation LAC Operation Whitecoat Project 112 San Jose Project == Notes == == References == == External links == Columbia Journalism Review, Laurels November/December 2000 Vietnam Veterans of America - The Veteran December2000/January 2001 http://archive.vva.org/TheVeteran/2002_01/hazardous.htm Project SHAD at the United States Department of Veterans Affairs, includes pocket guides and Q&A Vietnam Veterans of America GAO The Chemical-Biological Warfare Exposures Site, Official Web site of Force Health Protection & Readiness Policy & Programs Project 112/SHAD - Shipboard Hazard and Defense, Official Web site of Force Health Protection & Readiness Policy & Programs === Declassified documents === OSD & Joint Staff FOIA Requester Service Center PROJECT
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SHAD -- AUTUMN GOLD, May 1964 PROJECT SHAD -- BIG JACK PROJECT SHAD -- NIGHT TRAIN, December 1964
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The copula linguae or copula, is a swelling that forms from the second pharyngeal arch, late in the fourth week of embryogenesis. During the fifth and sixth weeks the copula becomes overgrown and covered by the hypopharyngeal eminence which forms mostly from the third pharyngeal arch and in part from the fourth pharyngeal arch. == References == This article incorporates text in the public domain from page 1103 of the 20th edition of Gray's Anatomy (1918) == External links == hednk-024—Embryo Images at University of North Carolina
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"title": "Copula linguae"
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Fungal DNA barcoding is the process of identifying species of the biological kingdom Fungi through the amplification and sequencing of specific DNA sequences and their comparison with sequences deposited in a DNA barcode database such as the ISHAM reference database, or the Barcode of Life Data System (BOLD). In this attempt, DNA barcoding relies on universal genes that are ideally present in all fungi with the same degree of sequence variation. The interspecific variation, i.e., the variation between species, in the chosen DNA barcode gene should exceed the intraspecific (within-species) variation. A fundamental problem in fungal systematics is the existence of teleomorphic and anamorphic stages in their life cycles. These morphs usually differ drastically in their phenotypic appearance, preventing a straightforward association of the asexual anamorph with the sexual teleomorph. Moreover, fungal species can comprise multiple strains that can vary in their morphology or in traits such as carbon- and nitrogen utilisation, which has often led to their description as different species, eventually producing long lists of synonyms. Fungal DNA barcoding can help to identify and associate anamorphic and teleomorphic stages of fungi, and through that to reduce the confusing multitude of fungus names. For this reason, mycologists were among the first to spearhead the investigation of species discrimination by means of DNA sequences, at least 10 years earlier than the DNA barcoding proposal for animals by Paul D. N. Hebert and colleagues in 2003, who popularised the term "DNA barcoding". The success of identification of fungi by means of DNA barcode sequences stands and falls with the quantitative (completeness) and qualitative (level of identification) aspect of the reference database. Without a database covering a broad taxonomic range of fungi, many identification queries will not result in a satisfyingly close match. Likewise, without a substantial curatorial effort to maintain the
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{
"page_id": 62065916,
"source": null,
"title": "Fungal DNA barcoding"
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records at a high taxonomic level of identification, queries – even when they might have a close or exact match in the reference database – will not be informative if the closest match is only identified to phylum or class level. Another crucial prerequisite for DNA barcoding is the ability to unambiguously trace the provenance of DNA barcode data back to the originally sampled specimen, the so-called voucher specimen. This is common practice in biology along with the description of new taxa, where the voucher specimens, on which the taxonomic description is based, become the type specimens. When the identity of a certain taxon (or a genetic sequence in the case of DNA barcoding) is in doubt, the original specimen can be re-examined to review and ideally solve the issue. Voucher specimens should be clearly labelled as such, including a permanent voucher identifier that unambiguously connects the specimen with the DNA barcode data derived from it. Furthermore, these voucher specimens should be deposited in publicly accessible repositories like scientific collections or herbaria to preserve them for future reference and to facilitate research involving the deposited specimens. == Barcode DNA markers == === Internal Transcribed Spacer (ITS) – the primary fungal barcode === In fungi, the Internal transcribed spacer (ITS) is a roughly 600 base pairs long region in the ribosomal tandem repeat gene cluster of the nuclear genome. The region is flanked by the DNA sequences for the ribosomal small subunit (SSU) or 18S subunit at the 5' end, and by the large subunit (LSU) or 28S subunit at the 3' end. The Internal Transcribed Spacer itself consists of two parts, ITS1 and ITS2, which are separated from each other by the 5.8S subunit nested between them. Like the flanking 18S and 28S subunits, the 5.8S subunit contains a highly
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{
"page_id": 62065916,
"source": null,
"title": "Fungal DNA barcoding"
}
|
conserved DNA sequence, as they code for structural parts of the ribosome, which is a key component in intracellular protein synthesis. Due to several advantages of ITS (see below) and a comprehensive amount of sequence data accumulated in the 1990s and early 2000s, Begerow et al. (2010) and Schoch et al. (2012) proposed the ITS region as primary DNA barcode region for the genetic identification of fungi. UNITE is an open ITS barcoding database for fungi and all other eukaryotes. ==== Primers ==== The conserved flanking regions of 18S and 28S serve as anchor points for the primers used for PCR amplification of the ITS region. Moreover, the conserved nested 5.8S region allows for the construction of "internal" primers, i.e., primers attaching to complementary sequences within the ITS region. White et al. (1990) proposed such internal primers, named ITS2 and ITS3, along with the flanking primers ITS1 and ITS4 in the 18S and the 28S subunit, respectively. Due to their almost universal applicability to ITS sequencing in fungi, these primers are still in wide use today. Optimised primers specifically for ITS sequencing in Dikarya (comprising Basidiomycota and Ascomycota) have been proposed by Toju et al. (2012). For the majority of fungi, the ITS primers proposed by White et al. (1990) have become the standard primers used for PCR amplification. These primers are: ==== Advantages and shortcomings ==== A major advantage of using the ITS region as molecular marker and fungal DNA barcode is that the entire ribosomal gene cluster is arranged in tandem repeats, i.e., in multiple copies. This allows for its PCR amplification and Sanger sequencing even from small material samples (given the DNA is not fragmented due to age or other degenerative influences). Hence, a high PCR success rate is usually observed when amplifying ITS. However, this success
|
{
"page_id": 62065916,
"source": null,
"title": "Fungal DNA barcoding"
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|
rate varies greatly among fungal groups, from 65% in non-Dikarya (including the now paraphyletic Mucoromycotina, the Chytridiomycota and the Blastocladiomycota) to 100% in Saccharomycotina and Basidiomycota (with the exception of very low success in Pucciniomycotina). Furthermore, the choice of primers for ITS amplification can introduce biases towards certain taxonomic fungus groups. For example, the "universal" ITS primers fail to amplify about 10% of the tested fungal specimens. The tandem repeats of the ribosomal gene cluster cause the problem of significant intragenomic sequence heterogeneity observed among ITS copies of several fungal groups. In Sanger sequencing, this will cause ITS sequence reads of different lengths to superpose each other, potentially rendering the resulting chromatograph unreadable. Furthermore, because of the non-coding nature of the ITS region that can lead to a substantial amount of indels, it is impossible to consistently align ITS sequences from highly divergent species for further bigger-scale phylogenetic analyses. The degree of intragenomic sequence heterogeneity can be investigated in more detail through molecular cloning of the initially PCR-amplified ITS sequences, followed by sequencing of the clones. This procedure of initial PCR amplification, followed by cloning of the amplicons and finally sequencing of the cloned PCR products is the most common approach of obtaining ITS sequences for DNA metabarcoding of environmental samples, in which a multitude of different fungal species can be present simultaneously. However, this approach of sequencing after cloning was rarely done for the ITS sequences that make up the reference libraries used for DNA barcode-aided identification, thus potentially giving an underestimate of the existing ITS sequence variation in many samples. The weighted arithmetic mean of the intraspecific (within-species) ITS variability among fungi is 2.51%. This variability, however, can range from 0% for example in Serpula lacrymans (n=93 samples) over 0.19% in Tuber melanosporum (n=179) up to 15.72% in
|
{
"page_id": 62065916,
"source": null,
"title": "Fungal DNA barcoding"
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Rhizoctonia solani (n=608), or even 24.75% in Pisolithus tinctorius (n=113). In cases of high intraspecific ITS variability, the application of a threshold of 3% sequence variability – a canonical upper value for intraspecific variation – will therefore lead to a higher estimate of operational taxonomic units (OTUs), i.e., putative species, than there actually are in a sample. In the case of medically relevant fungal species, a more strict threshold of 2.5% ITS variability allows only around 75% of all species to be accurately identified to the species level. On the other hand, morphologically well-defined, but evolutionarily young species complexes or sibling species may only differ (if at all) in a few nucleotides of the ITS sequences. Solely relying on ITS barcode data for the identification of such species pairs or complexes may thus obscure the actual diversity and might lead to misidentification if not accompanied by the investigation of morphological and ecological features and/or comparison of additional diagnostic genetic markers. For some taxa, ITS (or its ITS2 part) is not variable enough as fungal DNA barcode, as for example has been shown in Aspergillus, Cladosporium, Fusarium and Penicillium. Efforts to define a universally applicable threshold value of ITS variability that demarcates intraspecific from interspecific (between-species) variability thus remain futile. Nonetheless, the probability of correct species identification with the ITS region is high in the Dikarya, and especially so in Basidiomycota, where even the ITS1 part is often sufficient to identify the species. However, its discrimination power is partly superseded by that of the DNA-directed RNA polymerase II subunit RPB1 (see also below). Due to the shortcomings of ITS' as primary fungal DNA barcode, the necessity of establishing a second DNA barcode marker was expressed. Several attempts were made to establish other genetic markers that could serve as additional DNA barcodes,
|
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"page_id": 62065916,
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"title": "Fungal DNA barcoding"
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similar to the situation in plants, where the plastidial genes rbcL, matK and trnH-psbA, as well as the nuclear ITS are often used in combination for DNA barcoding. === Translational elongation factor 1α (TEF1α) – the secondary fungal barcode === The translational elongation factor 1α is part of the eucaryotic elongation factor 1 complex, whose main function is to facilitate the elongation of the amino acid chain of a polypeptide during the translation process of gene expression. Stielow et al. (2015) investigated the TEF1α gene, among a number of others, as potential genetic marker for fungal DNA barcoding. The TEF1α gene coding for the translational elongation factor 1α is generally considered to have a slow mutation rate, and it is therefore generally better suited for investigating older splits deeper in the phylogenetic history of an organism group. Despite this, the authors conclude that TEF1α is the most promising candidate for an additional DNA barcode marker in fungi as it also features sequence regions of higher mutation rates. Following this, a quality-controlled reference database was established and merged with the previously existing ISHAM-ITS database for fungal ITS DNA barcodes to form the ISHAM database. TEF1α has been successfully used to identify a new species of Cantharellus from Texas and distinguish it from a morphologically similar species. In the genera Ochroconis and Verruconis (Sympoventuriaceae, Venturiales), however, the marker does not allow distinction of all species. TEF1α has also been used in phylogenetic analyses at the genus level, e.g. in the case of Cantharellus and the entomopathogenic Beauveria, and for the phylogenetics of early-diverging fungal lineages. ==== Primers ==== TEF1α primers used in the broad-scale screening of the performance of DNA barcode gene candidates of Stielow et al. (2015) were the forward primer EF1-983F with the sequence 5'-GCYCCYGGHCAYCGTGAYTTYAT-3', and the reverse primer EF1-1567R
|
{
"page_id": 62065916,
"source": null,
"title": "Fungal DNA barcoding"
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|
with the sequence 5'-ACHGTRCCRATACCACCRATCTT-3'. In addition, a number of new primers was developed, with the primer pair in bold resulting in a high average amplification success of 88%: Primers used for the investigation of Rhizophydiales and especially Batrachochytrium dendrobatidis, a pathogen of amphibia, are the forward primer tef1F with the nucleotide sequence 5'-TACAARTGYGGTGGTATYGACA-3', and the reverse primer tef1R with the sequence 5'-ACNGACTTGACYTCAGTRGT-3'. These primers also successfully amplified the majority of Cantharellus species investigated by Buyck et al. (2014), with the exception of a few species for which more specific primers were developed: the forward primer tef-1Fcanth with the sequence 5'-AGCATGGGTDCTYGACAAG-3', and the reverse primer tef-1Rcanth with the sequence 5'-CCAATYTTRTAYACATCYTGGAG-3'. === D1/D2 domain of the LSU ribosomal RNA === The D1/D2 domain is part of the nuclear large subunit (28S) ribosomal RNA, and it is therefore located in the same ribosomal tandem repeat gene cluster as the Internal Transcribed Spacer (ITS). But unlike the non-coding ITS sequences, the D1/D2 domain contains coding sequence. With about 600 base pairs it is about the same nucleotide sequence length as ITS, which makes amplification and sequencing rather straightforward, an advantage that has led to the accumulation of an extensive amount of D1/D2 sequence data especially for yeasts. Regarding the molecular identification of basidiomycetous yeasts, D1/D2 (or ITS) can be used alone. However, Fell et al. (2000) and Scorzetti et al. (2002) recommend the combined analysis of the D1/D2 and ITS regions, a practice that later became the standard required information for describing new taxa of asco- and basidiomycetous yeasts. When attempting to identify early diverging fungal lineages, the study of Schoch et al. (2012), comparing the identification performance of different genetic markers, showed that the large subunit (as well as the small subunit) of the ribosomal RNA performs better than ITS or RPB1. ====
|
{
"page_id": 62065916,
"source": null,
"title": "Fungal DNA barcoding"
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|
Primers ==== For basidiomycetous yeasts, the forward primer F63 with the sequence 5'-GCATATCAATAAGCGGAGGAAAAG-3', and the reverse primer LR3 with the sequence 5'-GGTCCGTGTTTCAAGACGG-3' have been successfully used for PCR amplification of the D1/D23 domain. The D1/D2 domain of ascomycetous yeasts like Candida can be amplified with the forward primer NL-1 (same as F63) and the reverse primer NL-4 (same as LR3). === RNA polymerase II subunit RPB1 === The RNA polymerase II subunit RPB1 is the largest subunit of the RNA polymerase II. In Saccharomyces cerevisiae, it is encoded by the RPO21 gene. PCR amplification success of RPB1 is very taxon-dependent, ranging from 70 to 80% in Ascomycota to 14% in early diverging fungal lineages. Apart from the early diverging lineages, RPB1 has a high rate of species identification in all fungal groups. In the species-rich Pezizomycotina it even outperforms ITS. In a study comparing the identification performance of four genes, RPB1 was among the most effective genes when combining two genes in the analysis: combined analysis with either ITS or with the large subunit ribosomal RNA yielded the highest identification success. Other studies also used RPB2, the second-largest subunit of the RNA polymerase II, e.g. for studying the phylogenetic relationships among species of the genus Cantharellus or for a phylogenetic study shedding light on the relationships among early-diverging lineages in the fungal kingdom. ==== Primers ==== Primers successfully amplifying RPB1 especially in Ascomycota are the forward primer RPB1-Af with the sequence 5'-GARTGYCCDGGDCAYTTYGG-3', and the reverse primer RPB1-Ac-RPB1-Cr with the sequence 5'-CCNGCDATNTCRTTRTCCATRTA-3'. === Intergenic Spacer (IGS) of ribosomal RNA genes === The Intergenic Spacer (IGS) is the region of non-coding DNA between individual tandem repeats of the ribosomal gene cluster in the nuclear genome, as opposed to the Internal Transcribed Spacer (ITS) that is situated within these tandem repeats. IGS has
|
{
"page_id": 62065916,
"source": null,
"title": "Fungal DNA barcoding"
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|
been successfully used for the differentiation of strains of Xanthophyllomyces dendrorhous as well as for species distinction in the psychrophilic genus Mrakia (Cystofilobasidiales). Due to these results, IGS has been recommended as a genetic marker for additional differentiation (along with D1/D2 and ITS) of closely related species and even strains within one species in basidiomycete yeasts. The recent discovery of additional non-coding RNA genes in the IGS region of some basidiomycetes cautions against uncritical use of IGS sequences for DNA barcoding and phylogenetic purposes. === Other genetic markers === The cytochrome c oxidase subunit I (COI) gene outperforms ITS in DNA barcoding of Penicillium (Ascomycota) species, with species-specific barcodes for 66% of the investigated species versus 25% in the case of ITS. Furthermore, a part of the β-Tubulin A (BenA) gene exhibits a higher taxonomic resolution in distinguishing Penicillium species as compared to COI and ITS. In the closely related Aspergillus niger complex, however, COI is not variable enough for species discrimination. In Fusarium, COI exhibits paralogues in many cases, and homologous copies are not variable enough to distinguish species. COI also performs poorly in the identification of basidiomycote rusts of the order Pucciniales due to the presence of introns. Even when the obstacle of introns is overcome, ITS and the LSU rRNA (28S) outperform COI as DNA barcode marker. In the subdivision Agaricomycotina, PCR amplification success was poor for COI, even with multiple primer combinations. Successfully sequenced COI samples also included introns and possible paralogous copies, as reported for Fusarium. Agaricus bisporus was found to contain up to 19 introns, making the COI gene of this species the longest recorded, with 29,902 nucleotides. Apart from the substantial troubles of sequencing COI, COI and ITS generally perform equally well in distinguishing basidiomycote mushrooms. Topoisomerase I (TOP1) was investigated as additional
|
{
"page_id": 62065916,
"source": null,
"title": "Fungal DNA barcoding"
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|
DNA barcode candidate by Lewis et al. (2011) based on proteome data, with the developed universal primer pair being subsequently tested on actual samples by Stielow et al. (2015). The forward primer TOP1_501-F with the sequence 5'-TGTAAAACGACGGCCAGT-ACGAT-ACTGCCAAGGTTTTCCGTACHTACAACGC-3' (where the first section marks the universal M13 forward primer tail, the second part consisting of ACGAT a spacer, and the third part the actual primer) and reverse the primer TOP1_501-R with 5'-CAGGAAACAGCTATGA-CCCAGTCCTCGTCAACWGACTTRATRGCCCA-3' (the first section marking the universal M13 reverse primer tail, the second part the actual TOP1 reverse primer) amplify a fragment of approximately 800 base pairs. TOP1 was found to be a promising DNA barcode candidate marker for ascomycetes, where it can distinguish species in Fusarium and Penicillium – genera, in which the primary ITS barcode performs poorly. However, poor amplification success with the TOP1 universal primers is observed in early-diverging fungal lineages and basidiomycetes except Pucciniomycotina (where ITS PCR success is poor). Like TOP1, the Phosphoglycerate kinase (PGK) was among the genetic markers investigated by Lewis et al. (2011) and Stielow et al. (2015) as potential additional fungal DNA barcodes. A number of universal primers was developed, with the PGK533 primer pair, amplifying a circa 1,000 base pair fragment, being the most successful in most fungi except Basidiomycetes. Like TOP1, PGK is superior to ITS in species differentiation in ascomycete genera like Penicillium and Fusarium, and both PGK and TOP1 perform as good as TEF1α in distinguishing closely related species in these genera. == Applications == === Food safety === A citizen science project investigated the consensus between the labelling of dried, commercially sold mushrooms and the DNA barcoding results from these mushrooms. All samples were found to be correctly labelled. However, an obstacle was the unreliability of ITS reference databases in terms of the level of identification, as
|
{
"page_id": 62065916,
"source": null,
"title": "Fungal DNA barcoding"
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|
the two databases (GenBank and UNITE) used for ITS sequence comparison gave different identification results in some of the samples. Correct labelling of mushrooms intended for consumption was also investigated by Raja et al. (2016), who used the ITS region for DNA barcoding from dried mushrooms, mycelium powders, and dietary supplement capsules. In only 30% of the 33 samples did the product label correctly state the binomial fungus name. In another 30%, the genus name was correct, but the species epithet did not match, and in 15% of the cases not even the genus name of the binomial name given on the product label matched the result of the obtained ITS barcode. For the remaining 25% of the samples, no ITS sequence could be obtained. Xiang et al. (2013) showed that using ITS sequences, the commercially highly valuable the caterpillar fungus Ophiocordyceps sinensis and its counterfeit versions (O. nutans, O. robertsii, Cordyceps cicadae, C. gunnii, C. militaris, and the plant Ligularia hodgsonii) can be reliably identified to the species level. === Pathogenic fungi === A study by Vi Hoang et al. (2019) focused on the identification accuracy of pathogenic fungi using both the primary (ITS) and secondary (TEF1α) barcode markers. Their results show that in Diutina (a segregate of Candida) and Pichia, species identification is straightforward with either the ITS or the TEF1α as well as with a combination of both. In the Lodderomyces assemblage, which contains three of the five most common pathogenic Candida species (C. albicans, C. dubliniensis, and C. parapsilosis), ITS failed to distinguish Candida orthopsilosis and C. parapsilosis, which are part of the Candida parapsilosis complex of closely related species. TEF1α, on the other hand, allowed identification of all investigated species of the Lodderomyces clade. Similar results were obtained for Scedosporium species, which are attributed to
|
{
"page_id": 62065916,
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"title": "Fungal DNA barcoding"
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a wide range of localised to invasive diseases: ITS could not distinguish between S. apiospermum and S. boydii, whereas with TEF1α all investigated species of this genus could be accurately identified. This study therefore underlines the usefulness of applying more than one DNA barcoding marker for fungal species identification. === Conservation of cultural heritage === Fungal DNA barcoding has been successfully applied to the investigation of foxing phenomena, a major concern in the conservation of paper documents. Sequeira et al. (2019) sequenced ITS from foxing stains and found Chaetomium globosum, Ch. murorum, Ch. nigricolor, Chaetomium sp., Eurotium rubrum, Myxotrichum deflexum, Penicillium chrysogenum, P. citrinum, P. commune, Penicillium sp. and Stachybotrys chartarum to inhabit the investigated paper stains. Another study investigated fungi that act as biodeteriorating agents in the Old Cathedral of Coimbra, part of the University of Coimbra, a UNESCO World Heritage Site. Sequencing the ITS barcode of ten samples with classical Sanger as well as with Illumina next-generation sequencing techniques, they identified 49 fungal species. Aspergillus versicolor, Cladosporium cladosporioides, C. sphaerospermum, C. tenuissimum, Epicoccum nigrum, Parengyodontium album, Penicillium brevicompactum, P. crustosum, P. glabrum, Talaromyces amestolkiae and T. stollii were the most common species isolated from the samples. Another study concerning objects of cultural heritage investigated the fungal diversity on a canvas painting by Paula Rego using the ITS2 subregion of the ITS marker. Altogether, 387 OTUs (putative species) in 117 genera of 13 different classes of fungi were observed. == See also == DNA barcoding Microbial DNA barcoding Pollen DNA barcoding DNA barcoding in diet assessment Consortium for the Barcode of Life == References == == Further reading == == External links == Aftol primer listing (as used in James et al. 2006's six-gene phylogeny)
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{
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Charmed baryons are a category of composite particles comprising all baryons made of at least one charm quark. Since their first observation in the 1970s, a large number of distinct charmed baryon states have been identified. Observed charmed baryons have masses ranging between 2300 and 2700 MeV/c2. In 2002, the SELEX collaboration, based at Fermilab published evidence of a doubly charmed baryon (Ξ+cc), containing two charm quarks, with a mass of ~3520 MeV/c2, but has yet to be confirmed by other experiments. However, the LHCb collaboration claims to have found evidence for the doubly charmed baryon (Ξ++cc) in 2017. One triply charmed baryon (Ω++ccc) has been predicted but not yet observed. == Nomenclature == The nomenclature of charmed baryons is based on both quark content and isospin. The naming follows the rules established by the Particle Data Group. Charmed baryons composed of one charm quark and two up, one up and one down, or two down quarks are known as charmed Lambdas (Λc, isospin 0), or charmed Sigmas (Σc, isospin 1). Charmed baryons of isospin composed of one charm quark, and one up or down quark are known as charmed Xis (Ξc) and all have isospin 1/2. Charmed baryons composed of one charm quark and no up or down quarks are called charmed Omegas (Ωc) and all have isospin 0. Charmed baryons composed of two charm quarks and one up or down quark are called double charmed Xis (Ξcc) and all have isospin 1/2). Charmed baryons composed of two charm quarks and no up or down quarks are called double charmed Omega (Ωcc) and all have isospin 0. Charmed baryons composed of three charm quarks are called triple charmed Omegas (Ωccc), and all have isospin 0. Charge is indicated with superscripts. Heavy quark (bottom, charm, or top quarks) content is
|
{
"page_id": 3214588,
"source": null,
"title": "Charmed baryon"
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|
indicated by subscripts. For example, a Ξ+cb is made of one bottom, one charmed quark, and it can be deduced from the charge of the charm (+2/3e) and bottom quark (−1/3e) that the other quark must be an up quark (+2/3e). Sometimes asterisks or primes are used to indicate a resonance. == Properties == The important parameters of charmed baryons, to be studied, consist of four properties. They are firstly the mass, secondly the lifetime for those with a measurable lifetime, thirdly the intrinsic width (those particles that have too short a lifetime to measure have a measurable "width" or spread in mass due to Heisenberg's uncertainty principle), and lastly their decay modes. Compilations of measurements of these may be found in the publications of the Particle Data Group. == Production and detection == Charmed baryons are formed in high-energy particle collisions, such as those produced by particle accelerators. The general method to find them is to detect their decay products, identify what particles they are, and measure their momenta. If all the decay products are found and measured correctly, then the mass of the parent particle may be calculated. As an example, a favored decay of the Λ+c is into a proton, a kaon and a pion. The momenta of these (rather stable) particles are measured by the detector and using the usual rules of four-momentum using the correct relativistic equations, this gives a measure of the mass of the parent particle. In particle collisions, the protons, kaons and pions are all rather commonly produced, and only a fraction of these combinations will have come from a charmed baryon. Thus, it is important to measure many such combinations. A plot of the calculated parent mass will then have a peak at the mass of the Λ+c, but this is
|
{
"page_id": 3214588,
"source": null,
"title": "Charmed baryon"
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in addition to a smooth "phase space" background. The width of the peak will be governed by the resolution of the detector, provided that the charmed baryon is reasonably stable (such as the Λ+c which has a lifetime of around (2±10)×10−13 s). Other, higher states of charmed baryons, which decay by the strong interaction, typically have large intrinsic widths. This makes the peak stand up less definitively against the background combinations. First observations of particles by this method are notoriously difficult – overzealous interpretation of statistical fluctuations or effects that produce false "peaks" mean that several published results were later found to be false. However, with more data collected by more experiments over the years, the spectroscopy of the charmed baryons states has now reached a mature level. == Charmed lambda+ history == The first charmed baryon to be discovered was the Λ+c. It is not entirely clear when the particle was first observed; there were a number of experiments which published evidence for the state beginning in 1975, but the reported masses were frequently lower than the value now known. Since then, Λ+c have been produced and studied at many experiments, notably fixed-target experiments (such as FOCUS and SELEX) and e−e+ B-factories (ARGUS, CLEO, BABAR, and BELLE). === Mass === The definitive mass measurement has been made by the BaBar experiment, which reported a mass of 2286.46 MeV/c2 with a small uncertainty. To put this in context, it is more than twice as heavy as a proton. The excess mass is easily explained by the large constituent mass of the charm quark, which by itself is more than that of the proton. === Lifetime === The lifetime of the Λ+c is measured to be almost exactly 0.2 picoseconds. This is a typical lifetime for particles that decay via the
|
{
"page_id": 3214588,
"source": null,
"title": "Charmed baryon"
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weak interaction, taking into account the large available phase space. The lifetime measurement has contributions from a number of experiments, notably FOCUS, SELEX and CLEO. === Decays === The Λ+c decays into a multitude of different final states, according to the rules of weak decays. The decay into a proton, kaon and pion (each of them charged) is a favorite with experimenters as it is particularly easy to detect. It accounts for around 5% of all decays; around 30 distinct decay modes have been measured. Studies of these branching ratios enable theoreticians to disentangle the various fundamental diagrams contributing the decays and is a window on weak interaction physics. === Orbital excitations === The quark model, together with quantum mechanics predicts that there should be orbital excitations of Λ+c particles. The lowest lying of these states are ones where the two light quarks (the up and down) combine into a spin-0 state, one unit of orbital angular momentum is added, and this combines with the intrinsic spin of the charm quark to make a 1/2, 3/2 pair of particles. The higher of these (the Λ+c(2625)) was discovered in 1993 by ARGUS. At first it was not clear what state had been discovered, but the subsequent discovery of the lower state (2593) by CLEO clarified the situation. The decay modes, the masses, the measured widths, and the decays via two charged pions rather than one charged and one neutral pion, all confirm the identification of the states. == Charmed Sigma quark content == As noted above, charmed Sigma particles, like Λ+c particles, comprise a charm quark and two light up, down, strange) quarks. However, Σc particles have isospin 1. This is equivalent to saying that they can exist in three charged states, the doubly charged, the singly charged, and the neutral.
|
{
"page_id": 3214588,
"source": null,
"title": "Charmed baryon"
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|
The situation is directly analogous to the strange baryon nomenclature. The ground state (that is, with no orbital angular momentum) baryons can also be pictured thus. Each quark is a spin 1/2 particle. The spins can be pointed up, or down. In Λ+c ground state, the two light quarks point up-down to give a zero spin diquark. This then combines with the charm quark to give a spin 1/2 particle. In the Σc, the two light quarks combine to give a spin 1 diquark, which then combines with the charm quark to give either a spin 1/2 particle, or a spin 3/2 particle (normally known as a Σ∗c). It is the rules of quantum mechanics that make it possible for a Λc to exist only with three different quarks (that is cud quarks), whereas the Σc can exist as cuu, cud or cdd (thus the three different charges). All Σc particles decay by the strong force. Typically this mean the emission of a pion as it decays down to the comparatively stable Λ+c. Thus their masses are not usually measured directly, but in terms of their mass differences, m(Σc)−m(Λ+c). This is experimentally easier to measure precisely, and theoretically easier to predict, than the absolute value of the mass. === Σc(2455) history and mass === The lowest-mass Σc was given the name "2455" by the Particle Data Group, using their convention that strongly decaying particles are known by a rough value of their mass. It was searched for since the early days of charmed baryon studies. Individual events in bubble chambers were several times touted by experiments as evidence of the particles, but it is unclear how one event of this sort can be used as evidence of a resonance. As early as 1979, there was reasonable evidence of the doubly
|
{
"page_id": 3214588,
"source": null,
"title": "Charmed baryon"
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|
charged state from the Columbia–Brookhaven collaboration. In 1987–89, a series of experiments (E-400 at Fermilab, ARGUS and CLEO) with much larger statistics, found clear evidence for both the doubly charged and neutral states (though the E-400 neutral state turned out to be a false signal). It became clear that the mass difference m(Σc) − m(Λ+c) is around 168 MeV/c2. The singly-charged state was harder to detect, not because it is harder to produce, but simply because its decay via a neutral pion has more background and inferior resolution when detected by most particle detectors. It was not found (except for the report of a single event), until 1993 by CLEO. The intrinsic width of the Σc is small by the standard of most strong decays, but has now been measured, at least for the neutral and doubly charged states, to be around 2 MeV/c2 by the CLEO and FOCUS detectors. The next state up in mass is the spin 3/2 state, usually known as the Σ∗c or the Σc(2520). These are clearly going to be "wider" because of the extra phase space of their decay, which like the Σc(2455) is to one pion plus a ground-state Λc. Again, large statistics are necessary to claim a signal above the large number of Λc-π pairs that are produced. Again, the neutral and doubly charged states are experimentally easier to detect, and these were discovered in 1997 by the CLEO Collaboration. The singly charged state had to wait till 2001 by the time they had collected more data. == Ξc history and mass == In the standard quark model, Ξ+c comprises a csu quark combination and the Ξ0c comprises a csd quark combination. Both particles decay via the weak interaction. The first observation of the Ξ+c was in 1983 by the WA62 collaboration
|
{
"page_id": 3214588,
"source": null,
"title": "Charmed baryon"
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|
working at CERN. They found a significant peak in the decay mode ΛK−π+π+ at a mass of 2460±25 MeV/c2. The present value for the mass is taken from an average of 6 experiments, and is 2467.9±0.4 MeV/c2. The Ξ0c was discovered in 1989 by the CLEO, who measured a peak in the decay mode of Ξ−π+ with a mass of 2471±5 MeV/c2. The accepted value is 2471.0±0.4 MeV/c2. == Charmed Omega history and mass == Not surprisingly, of the four weakly decaying, singly charmed baryons, the Ωc (the css quark combination), was the last to be discovered and the least-well measured. Its history is murky. Some authors claim that in 1985 a cluster of three events observed at CERN was a signal, but this can now be excluded on the grounds of its incorrect mass. The ARGUS experiment published a small peak as a possible signal in 1993, but this can now be excluded on cross-section grounds, as many experiments have operated in the same environment as ARGUS with many more collisions. The E-687 experiment at Fermilab published two papers, one in 1993 and the other in 1994. The former one showed a small peak of marginal significance in the decay mode Ωπ, and a larger, apparently robust signal in the decay mode Σ+K−K−π+. This latter observation is considered valid by the Particle Data Group, but increasingly seems odd in that this decay mode has not been observed by other experiments. The CLEO experiment then showed a peak of 40 events in the sum of a variety of decay modes and a mass of 2494.6 MeV/c2. Since then, two experiments, BaBar, and Belle, have taken a great deal of data, and have shown very strong signals at a mass very similar to the CLEO value. However, neither have done the
|
{
"page_id": 3214588,
"source": null,
"title": "Charmed baryon"
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|
necessary studies to be able to quote a mass with an uncertainty. Therefore, though there is no doubt that the particle has been discovered, there is no definitive measurement of its mass. == References ==
|
{
"page_id": 3214588,
"source": null,
"title": "Charmed baryon"
}
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Lilabati Bhattacharjee (née Ray) was a mineralogist, crystallographer and a physicist. She studied with the scientist Satyendra Nath Bose, and completed her MSc in physics from the University College of Science and Technology (commonly known as Rajabazar Science College), University of Calcutta in 1951. Mrs Bhattacharjee worked in the fields of structural crystallography, optical transform methods, computer programming, phase transformations, crystal growth, topography, and instrumentation. She served as a Senior Mineralogist at the Geological Survey of India, and later went on to become its Director (Mineral Physics). She was married to Siva Brata Bhattacherjee, and is survived by two children and two grandsons. == References ==
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{
"page_id": 53349635,
"source": null,
"title": "Lilabati Bhattacharjee"
}
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A fruit press is a device used to separate fruit solids—stems, skins, seeds, pulp, leaves, and detritus—from fruit juice. == History == In the United States, Madeline Turner invented the Turner's Fruit-Press, in 1916. == Cider press == A cider press is used to crush apples or pears. In North America, the unfiltered juice is referred to as cider, becoming known as apple juice once filtered; in Britain it is referred to as juice regardless of whether it is filtered or not (the term cider is reserved for the fermented (alcoholic) juice). Other products include cider vinegar, (hard) cider, apple wine, apple brandy, and apple jack. The traditional cider press is a ram press. Apples are ground up and placed in a cylinder, and a piston exerts pressure. The cylinder and/or piston in a traditional cider press is designed to allow the juice to escape while retaining the solid matter. This is achieved through a controlled gap or a semi-permeable surface, where the pressure exerted by the piston forces the juice out of the crushed apples or pears, leaving the solids behind. The traditional cider press has not changed much since the early modern period. The only difference being that in earlier versions of the press horses were used to power the machine. Diderot's Encyclopedie offers a portrayal of the traditional cider press, "This is how the cider mill is made. Imagine a circular trough made of wood connected to two wooden millstones like those used in a windmill, but fixed differently. In a windmill, they are horizontal, but in the cider mill they are placed in the trough vertically. They are fixed to a vertical piece of wood that turns on itself and which is placed in the centre of the circular part of the trough; a long axle
|
{
"page_id": 68869,
"source": null,
"title": "Fruit press"
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passes through them; the axle is joined to the vertical axis; its other end juts out from the trough; a horse → is harnessed to it; the ← horse → pulls the axle by walking round the trough, which also moves the pressing stones in the trough where the apples are pounded. When they are judged to be sufficiently crushed, that is to say, enough for all the juice to be extracted from them, the apples are removed with a wooden spade and put into a large vat nearby. Enough apples are pounded to make a pulp or pomace." Cider presses often have attachments to grind the apples prior to pressing. Such combination devices are commonly referred to as cider mills. In communities with many small orchards, it is common for one or more persons to have a large cider mill for community use. These community mills allow orchard owners to avoid the capital, space, and maintenance requirements for having their own mill. These larger mills are typically powered by electrical or gasoline engines. Mill operators also deal with the solids, which attract wasps or hornets. Cider mills typically give patrons a choice between paying by the gallon/litre or splitting the cider with the mill operator. Larger orchardists may prefer to have their own presses because it saves on fees, or because it reduces cartage. Orchardists of any size may believe their own sanitation practices to be superior to that of community mills, as some patrons of community mills may make cider from low quality fruit (windfall apples, or apples with worms). Those making speciality ciders, such as pear cider, may want to have their own press. The world's largest cider press is located in Berne, Indiana, US. == Wine press == A wine press is a device used to
|
{
"page_id": 68869,
"source": null,
"title": "Fruit press"
}
|
press grapes during wine making. == Oil press == An oil press is a device used to extract oil from plants and fruits. == DIY fruit press == Given the simplicity of the design, and high usability, some people (e.g. those owning an orchard) have started building their own do-it-yourself (DIY) fruit press and have uploaded detailed instructions on how to do so. == See also == Cider mill == References == This article incorporates text from a publication now in the public domain: Easton, Matthew George (1897). Easton's Bible Dictionary (New and revised ed.). T. Nelson and Sons. {{cite encyclopedia}}: Missing or empty |title= (help) == External links == Mother Earth cider press plan
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{
"page_id": 68869,
"source": null,
"title": "Fruit press"
}
|
BioSentinel is a lowcost CubeSat spacecraft on a astrobiology mission that will use budding yeast to detect, measure, and compare the impact of deep space radiation on DNA repair over long time beyond low Earth orbit. Selected in 2013 for a 2022 launch, the spacecraft will operate in the deep space radiation environment throughout its 18-month mission. This will help scientists understand the health threat from cosmic rays and deep space environment on living organisms and reduce the risk associated with long-term human exploration, as NASA plans to send humans farther into space than ever before. The spacecraft was launched on 16 November 2022 as part of the Artemis 1 mission. In August 2023, NASA extended BioSentinel's mission into November 2024. The mission was developed by NASA Ames Research Center. == Background == BioSentinel is one of ten low-cost CubeSat missions that flew as secondary payloads aboard Artemis 1, the first test flight of NASA's Space Launch System. The spacecraft was deployed in cis-lunar space as NASA's first mission to send living organisms beyond low Earth orbit since Apollo 17 in 1972. == Objective == The primary objective of BioSentinel is to develop a biosensor using a simple model organism (yeast) to detect, measure, and correlate the impact of space radiation to living organisms over long durations beyond low Earth orbit (LEO) and into heliocentric orbit. While progress has been made with simulations, no terrestrial laboratory can duplicate the unique space radiation environment. == Biological science == The BioSentinel biosensor uses the budding yeast Saccharomyces cerevisiae to detect and measure DNA damage response after exposure to the deep space radiation environment. Two yeast strains were selected for this mission: a wild type strain proficient in DNA repair, and a strain defective in the repair of DNA double strand breaks (DSBs),
|
{
"page_id": 46796038,
"source": null,
"title": "BioSentinel"
}
|
deleterious lesions generated by ionizing radiation. Budding yeast was selected not only because of its flight heritage, but also because of its similarities with human cells, especially its DSB repair mechanisms. The biosensor consists of specifically engineered yeast strains and growth medium containing a metabolic indicator dye. Therefore, culture growth and metabolic activity of yeast cells directly indicate successful repair of DNA damage. After completing the Moon flyby and spacecraft checkout, the science mission phase will begin with the wetting of the first set of yeast-containing wells with specialized media. Multiple sets of wells will be activated at different time points over the 18-month mission. One reserve set of wells will be activated in the occurrence of a solar particle event (SPE). Approximately, a 4 to 5 krad total ionizing dose is anticipated. Payload science data and spacecraft telemetry will be stored on board and then downloaded to the ground. Biological measurements will be compared to data provided by onboard radiation sensors and dosimeters. Additionally, two identical BioSentinel payloads have been developed: one for the International Space Station (ISS), which is in similar microgravity conditions but a comparatively low-radiation environment, and one for use as a delayed-synchronous ground control at Earth gravity and, due to Earth's magnetic field, at Earth-surface-level radiation. The payload on the ISS has been warmed up and rehydrated in January 2022, the one on Earth surface, weeks later. They will help calibrate the biological effects of radiation in deep space to analogous measurements conducted on Earth and on the ISS. == Spacecraft == The Biosentinel spacecraft will consist of a 6U CubeSat bus format, with external dimensions of 10 cm × 20 cm × 30 cm and a mass of about 14 kg (31 lb). At launch, BioSentinel resides within the second stage on the launch
|
{
"page_id": 46796038,
"source": null,
"title": "BioSentinel"
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vehicle from which it is deployed to a lunar flyby trajectory and into an Earth-trailing heliocentric orbit. Of the total 6 Units volume, 4 Units will hold the science payload, including a radiation dosimeter and a dedicated 3-color spectrometer for each well; 0.5U will house the ADCS (Attitude Determination and Control Subsystem), 0.5U will house the EPS (Electrical Power System) and C&DH (Command and Data Handling) avionics, and 1U will house the attitude control thruster assembly, which will be 3D printed all in one piece: cold gas (DuPont R236fa) propellant tanks, lines and seven nozzles. The use of 3D printing also allows the optimization of space for increased propellant storage (165 grams ). The thrust of each nozzle is 50 mN, and a specific impulse of 31 seconds. The attitude control system is being developed and fabricated by the Georgia Institute of Technology. Electric power will be generated by deployable solar panels rated at 30 watts, and telecommunications will rely on the Iris transponder at X-band. The spacecraft is being developed by NASA Ames Research Center (AMR), in collaboration with NASA Jet Propulsion Laboratory (JPL), NASA Johnson Space Center (JSC), NASA Marshall Space Flight Center (MSFC), and NASA Headquarters. == See also == The 10 CubeSats flying in the Artemis 1 mission Near-Earth Asteroid Scout by NASA was a solar sail spacecraft that was planned to encounter a near-Earth asteroid (mission failure) BioSentinel is an astrobiology mission LunIR by Lockheed Martin Space Lunar IceCube, by the Morehead State University CubeSat for Solar Particles (CuSP) Lunar Polar Hydrogen Mapper (LunaH-Map), designed by the Arizona State University EQUULEUS, submitted by JAXA and the University of Tokyo OMOTENASHI, submitted by JAXA, was a lunar lander (mission failure) ArgoMoon, designed by Argotec and coordinated by Italian Space Agency (ASI) Team Miles, by Fluid and
|
{
"page_id": 46796038,
"source": null,
"title": "BioSentinel"
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|
Reason LLC, Tampa, Florida The 3 CubeSat missions removed from Artemis 1 Lunar Flashlight will map exposed water ice on the Moon Cislunar Explorers, Cornell University, Ithaca, New York Earth Escape Explorer (CU-E3), University of Colorado Boulder Astrobiology missions Bion BIOPAN Biosatellite program List of microorganisms tested in outer space O/OREOS OREOcube Tanpopo == References == == External links == Fact Sheet of BioSentinel, at NASA
|
{
"page_id": 46796038,
"source": null,
"title": "BioSentinel"
}
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In theoretical physics, the Pasterski–Strominger–Zhiboedov (PSZ) triangle or infrared triangle is a series of relationships between three groups of concepts involving the theory of relativity, quantum field theory and quantum gravity. The triangle highlights connections already known or demonstrated by its authors, Sabrina Gonzalez Pasterski, Andrew Strominger and Alexander Zhiboedov. The connections are among weak and lasting effects caused by the passage of gravitational or electromagnetic waves (memory effects), quantum field theorems on graviton and photon and geometrical symmetries of spacetime. Because all of this occurs under conditions of low energy, known as infrared in the language of physicists, it is also referred to as the infrared triangle. == Elements of the triangle == === Related concepts === The concepts that are interconnected by the triangle are: a) soft particle theorems (quantum field theory theorems regarding the behavior of low-energy gravitons or photons): soft graviton theorem, published by Steven Weinberg in 1965; extension of the previous theorem, published by Freddy Cachazo and Strominger in 2014; soft photon theorem, also published by Weinberg in the same paper of 1965 regarding the graviton; b) asymptotic symmetries (symmetries of spacetime distant from the sources of the fields): supertranslations of the Bondi-Metzner-Sachs group, published in 1962; superrotations (symmetry analogous to that of the Virasoro algebra), published by Glenn Barnich and Cédric Troessaert in 2010; symmetries of U(1) gauge theories, published by Pasterki in 2017; c) memory effects: gravitational memory effect, published by Yakov Zeldovich and A. G. Polnarev in 1974 and Demetrios Christodoulou in 1991; new gravitational memory effects, published by Pasterski, Strominger and Zhiboedov in 2016; the electromagnetic analogue of the memory effect, published by Lydia Bieri and David Garfinkle in 2013. === Binding relationships === Each group is linked to another by special relationships: Fourier transforms tie together soft theorems and memory
|
{
"page_id": 75500805,
"source": null,
"title": "Pasterski–Strominger–Zhiboedov triangle"
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|
effects; vacuum transitions tie together asymptotic symmetries and memory effects; Ward's identities tie together soft theorems and asymptotic symmetries. So, for example: the soft graviton theorem (a.1) is related to the supertranslations (b.1) by a Ward's identity; the supertranslations (b.1) correspond to different vacuum states created by the gravitational memory effect (c.1) the gravitational memory effect (c.1) reduces to the soft graviton theorem (a.1) via a Fourier transform. In addition to the first triangular relationship highlighted by the authors, several others may exist and have been hypothesized. == See also == Gravitational memory effect Bondi-Metzner-Sachs group Soft graviton theorem == References == == Bibliography == Andrew Strominger (2018). Lectures on the Infrared Structure of Gravity and Gauge Theory. Princeton University Press. == External links == Schwichtenberg, Jakob (16 October 2017). "Surprising Symmetries at the other End of the Spectrum". Jakob Schwichtenberg.
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{
"page_id": 75500805,
"source": null,
"title": "Pasterski–Strominger–Zhiboedov triangle"
}
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Foundations of the Czech chemical nomenclature (Czech: české chemické názvosloví) and terminology were laid during the 1820s and 1830s. These early naming conventions fit the Czech language and, being mostly the work of a single person, Jan Svatopluk Presl, provided a consistent way to name chemical compounds. Over time, the nomenclature expanded considerably, following the recommendations by the International Union of Pure and Applied Chemistry (IUPAC) in the recent era. Unlike the nomenclature that is used in biology or medicine, the chemical nomenclature stays closer to the Czech language and uses Czech pronunciation and inflection rules, but developed its own, very complex, system of morphemes (taken from Greek and Latin), grammar, syntax, punctuation and use of brackets and numerals. Certain terms (such as etanol 'ethanol') use the phonetic transcription, but the rules for spelling are inconsistent. == History == Medieval alchemists in the Czech lands used obscure and inconsistent terminology to describe their experiments. Edward Kelley, an alchemist at the court of Rudolf II, even invented his own secret language. Growth of the industry in the region during the 19th century, and the nationalistic fervour of the Czech National Revival, led to the development of Czech terminologies for natural and applied sciences. Jan Svatopluk Presl (1791–1849), an all-round natural scientist, proposed a new Czech nomenclature and terminology in the books Lučba čili chemie zkusná (1828–1835) and Nerostopis (1837). Presl had invented Czech neologisms for most of the then known chemical elements; ten of these, including vodík 'hydrogen', kyslík 'oxygen', uhlík 'carbon', dusík 'nitrogen' and křemík 'silicon', have entered the language. Presl also created naming conventions for oxides, in which the electronegative component of the compound became the noun and the electropositive component became an adjective. The adjectives were associated with a suffix, according to the valence number of the component
|
{
"page_id": 14617864,
"source": null,
"title": "Czech chemical nomenclature"
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|
they represented. Originally there were five suffixes: -ný, -natý, -itý, -ový, and -elý. These were later expanded to eight by Vojtěch Šafařík: -ný, -natý, -itý, -ičitý, -ičný and -ečný, -ový, -istý, and -ičelý, representing oxidation numbers from 1 to 8. For example, železnatý corresponds to 'ferrous' and železitý to 'ferric'. Salts were identified by the suffix -an added to the noun. Many of the terms created by Presl derive from Latin, German or Russian; only some were retained in use. A similar attempt published in Orbis pictus (1852) by Karel Slavoj Amerling (1807–1884) to create Czech names for the chemical elements (and to order the elements into a structure, similar to the work of Russian chemist Nikolay Beketov) was not successful. Later work on the nomenclature was performed by Vojtěch Šafařík (1829–1902). In 1876 Šafařík started to publish the journal Listy chemické, the first chemistry journal in Austria-Hungary (today issued under the name Chemické Listy), and this journal has played an important role in the codification of the nomenclature and terminology. During a congress of Czech chemists in 1914, the nomenclature was reworked, and the new system became normative in 1918. Alexandr Sommer-Batěk (1874–1944) and Emil Votoček (1872–1950) were the major proponents of this change. Presl's original conventions remained in use, but formed only a small part of the naming system. Several changes were applied to the basic terminology during the second half of the 20th century, usually moving closer to the international nomenclature. For example, the former term kysličník 'oxide' was officially replaced by oxid, uhlovodan 'carbohydrate' by sacharid and later even karbohydrát. The spelling of some chemical elements also changed: berylium 'beryllium' should now be written beryllium. Adoption of these changes by the Czech public has been quite slow, and the older terms are still used decades later.
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{
"page_id": 14617864,
"source": null,
"title": "Czech chemical nomenclature"
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|
The Czechoslovak Academy of Sciences, founded in 1953, took over responsibility for maintenance of the nomenclature and proper implementation of the IUPAC recommendations. Since the Velvet Revolution (1989) this activity has slowed down considerably. == Oxidation state suffixes == == Notes == == External links == Website about the early history of the Czech chemical nomenclature (in Czech) Article in a Czech Academy of Sciences bulletin: current problems faced by the Czech chemical nomenclature (2000, section "Současný stav a problémy českého chemického názvosloví") === Organizations === Journal Chemické listy (nomenclature related articles are in Czech, ISSN 1213-7103, printed version ISSN 0009-2770) Czech Chemical Society (Česká společnost chemická, ČSCH, founded in 1866) National IUPAC Centre for the Czech Republic
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{
"page_id": 14617864,
"source": null,
"title": "Czech chemical nomenclature"
}
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Arcanadea (アルカナディア, Arukanadia) is a Japanese series of heavily customizable plastic model kit girls created and produced by Kotobukiya, released since December 2021, with necömi as the character designer. An anime television series adaptation has been announced. == Characters == Lumitea (ルミティア, Rumitia) Voiced by: Kaede Hondo Velretta (ヴェルルッタ, Vuerurutta) Voiced by: Miho Okasaki Yukumo (ユクモ) Voiced by: Rina Hidaka Elena (エレーナ, Erēna) Voiced by: Fairouz Ai Soffiera (ソフィエラ, Sofiera) Voiced by: Ai Kakuma Ermeda (エルメダ, Erumeda) Voiced by: M.A.O Charmed (シャルメド, Sharumedo) == Other media == === Anime === An anime television series adaptation was announced during a livestream event for the plastic model series on December 20, 2024. It is set to premiere on TV Asahi. == See also == Alice Gear Aegis Busou Shinki Frame Arms Girl, another plastic model series created by Kotobukiya Hundred Infinite Stratos Little Battlers Experience Symphogear == References == == External links == Official website (in Japanese) Official anime website (in Japanese) Arcanadea (anime) at Anime News Network's encyclopedia
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{
"page_id": 78646537,
"source": null,
"title": "Arcanadea"
}
|
Congelation (from Latin: congelātiō, lit. 'freezing, congealing') was a term used in medieval and early modern alchemy for the process known today as crystallization. In the Secreta alchymiae ('The Secret of Alchemy') attributed to Khalid ibn Yazid (c. 668–704 or 709), it is one of "the four principal operations", along with Solution, Albification ('whitening'), and Rubification ('reddening'). It was one of the twelve alchemical operations involved in the creation of the philosophers' stone as described by Sir George Ripley (c. 1415–1490) in his Compound of Alchymy, as well as by Antoine-Joseph Pernety in his Dictionnaire mytho-hermétique (1758). == See also == Alchemical process Magnum opus (alchemy) == References == === Works cited === Holmyard, Eric J. (1957). Alchemy. Harmondsworth: Penguin Books. OCLC 2080637. Linden, Stanton J. (2003). The Alchemy Reader: From Hermes Trismegistus to Isaac Newton. Cambridge: Cambridge University Press. ISBN 0-521-79234-7.
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{
"page_id": 2428176,
"source": null,
"title": "Congelation"
}
|
Foxfire, also called fairy fire and chimpanzee fire, is the bioluminescence created by some species of fungi present in decaying wood. The bluish-green glow is attributed to a luciferase, an oxidative enzyme, which emits light as it reacts with a luciferin. The phenomenon has been known since ancient times, with its source determined in 1823. == Description == Foxfire is the bioluminescence created by some species of fungi present in decaying wood. It occurs in a number of species, including Panellus stipticus, Omphalotus olearius and Omphalotus nidiformis. The bluish-green glow is attributed to luciferin, which emits light after oxidation catalyzed by the enzyme luciferase. Some believe that the light attracts insects to spread spores, or acts as a warning to hungry animals, like the bright colors exhibited by some poisonous or unpalatable animal species. Although generally very dim, in some cases foxfire is bright enough to read by. == History == The oldest recorded documentation of foxfire is from 382 B.C., by Aristotle, whose notes refer to a light that, unlike fire, was cold to the touch. The Roman thinker Pliny the Elder also mentioned glowing wood in olive groves. Foxfire was used to illuminate the needles on the barometer and the compass of Turtle, an early submarine. This is commonly thought to have been suggested by Benjamin Franklin; a reading of the correspondence from Benjamin Gale, however, shows that Benjamin Franklin was only consulted for alternative forms of lighting when the cold temperatures rendered the foxfire inactive. After many more literary references to foxfire by early scientists and naturalists, its cause was discovered in 1823. The glow emitted from wooden support beams in mines was examined, and it was found that the luminescence came from fungal growth. The "fox" in foxfire may derive from the Old French word faux,
|
{
"page_id": 1248534,
"source": null,
"title": "Foxfire"
}
|
meaning "false", rather than from the name of the animal. The association of foxes with such lights is widespread, however, and occurs also in Japanese folklore in the form of kitsune-bi (狐火 lit. 'Fox's fire/flame'). == See also == Aurora Borealis, called "revontulet" (literally "foxfires") in the Finnish language List of bioluminescent fungi Will-o'-the-wisp == References == == External links == Foxfire: Bioluminescence in the Forest PDF file by Dr. Kim D. Coder, University of Georgia 8/99 Bioluminescent Fungi at Mykoweb
|
{
"page_id": 1248534,
"source": null,
"title": "Foxfire"
}
|
Giomar Helena Borrero-Pérez is a Colombian marine biologist. In 2012 she became the sixth Colombian scientist to be awarded a L'Oréal-UNESCO For Women in Science Award. Her work considers the conservation of sea cucumbers. == Early life and education == Borrero was born in Mitú. She attended the National Indigenous Normal School of Mitú until 1992. Borrero was awarded a scholarship from Ecopetrol to attend Jorge Tadeo Lozano University, where she studied marine biology. She moved to Spain for her doctoral studies, where she worked on biodiversity at the University of Murcia. Whilst in Spain she started to work on the Holothuroidea species. == Research and career == Borrero completed an internship at the José Benito Vives de Andréis Marine and Coastal Research Institute. She served as a curator at the Museum of Marine History in Colombia. Her research considers sea cucumbers from around the Colombian - Caribbean coast. In 2014 she joined the Smithsonian Tropical Research Institute where she worked as a postdoctoral researcher studying the connectivity of Isostichopus badionotus populations. === Awards and honours === 2003 - Young Researchers Award, Colciencias 2004 - European Commission AlBan Scholarship 2012 - L'Oréal-UNESCO For Women in Science Award == References ==
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{
"page_id": 62393626,
"source": null,
"title": "Giomar Helena Borrero-Pérez"
}
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Filters in topology, a subfield of mathematics, can be used to study topological spaces and define all basic topological notions such as convergence, continuity, compactness, and more. Filters, which are special families of subsets of some given set, also provide a common framework for defining various types of limits of functions such as limits from the left/right, to infinity, to a point or a set, and many others. Special types of filters called ultrafilters have many useful technical properties and they may often be used in place of arbitrary filters. Filters have generalizations called prefilters (also known as filter bases) and filter subbases, all of which appear naturally and repeatedly throughout topology. Examples include neighborhood filters/bases/subbases and uniformities. Every filter is a prefilter and both are filter subbases. Every prefilter and filter subbase is contained in a unique smallest filter, which they are said to generate. This establishes a relationship between filters and prefilters that may often be exploited to allow one to use whichever of these two notions is more technically convenient. There is a certain preorder on families of sets (subordination), denoted by ≤ , {\displaystyle \,\leq ,\,} that helps to determine exactly when and how one notion (filter, prefilter, etc.) can or cannot be used in place of another. This preorder's importance is amplified by the fact that it also defines the notion of filter convergence, where by definition, a filter (or prefilter) B {\displaystyle {\mathcal {B}}} converges to a point if and only if N ≤ B , {\displaystyle {\mathcal {N}}\leq {\mathcal {B}},} where N {\displaystyle {\mathcal {N}}} is that point's neighborhood filter. Consequently, subordination also plays an important role in many concepts that are related to convergence, such as cluster points and limits of functions. In addition, the relation S ≥ B , {\displaystyle {\mathcal
|
{
"page_id": 47516955,
"source": null,
"title": "Filters in topology"
}
|
{S}}\geq {\mathcal {B}},} which denotes B ≤ S {\displaystyle {\mathcal {B}}\leq {\mathcal {S}}} and is expressed by saying that S {\displaystyle {\mathcal {S}}} is subordinate to B , {\displaystyle {\mathcal {B}},} also establishes a relationship in which S {\displaystyle {\mathcal {S}}} is to B {\displaystyle {\mathcal {B}}} as a subsequence is to a sequence (that is, the relation ≥ , {\displaystyle \geq ,} which is called subordination, is for filters the analog of "is a subsequence of"). Filters were introduced by Henri Cartan in 1937 and subsequently used by Bourbaki in their book Topologie Générale as an alternative to the similar notion of a net developed in 1922 by E. H. Moore and H. L. Smith. Filters can also be used to characterize the notions of sequence and net convergence. But unlike sequence and net convergence, filter convergence is defined entirely in terms of subsets of the topological space X {\displaystyle X} and so it provides a notion of convergence that is completely intrinsic to the topological space; indeed, the category of topological spaces can be equivalently defined entirely in terms of filters. Every net induces a canonical filter and dually, every filter induces a canonical net, where this induced net (resp. induced filter) converges to a point if and only if the same is true of the original filter (resp. net). This characterization also holds for many other definitions such as cluster points. These relationships make it possible to switch between filters and nets, and they often also allow one to choose whichever of these two notions (filter or net) is more convenient for the problem at hand. However, assuming that "subnet" is defined using either of its most popular definitions (which are those given by Willard and by Kelley), then in general, this relationship does not extend to
|
{
"page_id": 47516955,
"source": null,
"title": "Filters in topology"
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subordinate filters and subnets because as detailed below, there exist subordinate filters whose filter/subordinate-filter relationship cannot be described in terms of the corresponding net/subnet relationship; this issue can however be resolved by using a less commonly encountered definition of "subnet", which is that of an AA-subnet. Thus filters/prefilters and this single preorder ≤ {\displaystyle \,\leq \,} provide a framework that seamlessly ties together fundamental topological concepts such as topological spaces (via neighborhood filters), neighborhood bases, convergence, various limits of functions, continuity, compactness, sequences (via sequential filters), the filter equivalent of "subsequence" (subordination), uniform spaces, and more; concepts that otherwise seem relatively disparate and whose relationships are less clear. == Motivation == Archetypical example of a filter The archetypical example of a filter is the neighborhood filter N ( x ) {\displaystyle {\mathcal {N}}(x)} at a point x {\displaystyle x} in a topological space ( X , τ ) , {\displaystyle (X,\tau ),} which is the family of sets consisting of all neighborhoods of x . {\displaystyle x.} By definition, a neighborhood of some given point x {\displaystyle x} is any subset B ⊆ X {\displaystyle B\subseteq X} whose topological interior contains this point; that is, such that x ∈ Int X B . {\displaystyle x\in \operatorname {Int} _{X}B.} Importantly, neighborhoods are not required to be open sets; those are called open neighborhoods. Listed below are those fundamental properties of neighborhood filters that ultimately became the definition of a "filter." A filter on X {\displaystyle X} is a set B {\displaystyle {\mathcal {B}}} of subsets of X {\displaystyle X} that satisfies all of the following conditions: Not empty: X ∈ B {\displaystyle X\in {\mathcal {B}}} – just as X ∈ N ( x ) , {\displaystyle X\in {\mathcal {N}}(x),} since X {\displaystyle X} is always a neighborhood of x {\displaystyle
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{
"page_id": 47516955,
"source": null,
"title": "Filters in topology"
}
|
x} (and of anything else that it contains); Does not contain the empty set: ∅ ∉ B {\displaystyle \varnothing \not \in {\mathcal {B}}} – just as no neighborhood of x {\displaystyle x} is empty; Closed under finite intersections: If B , C ∈ B then B ∩ C ∈ B {\displaystyle B,C\in {\mathcal {B}}{\text{ then }}B\cap C\in {\mathcal {B}}} – just as the intersection of any two neighborhoods of x {\displaystyle x} is again a neighborhood of x {\displaystyle x} ; Upward closed: If B ∈ B and B ⊆ S ⊆ X {\displaystyle B\in {\mathcal {B}}{\text{ and }}B\subseteq S\subseteq X} then S ∈ B {\displaystyle S\in {\mathcal {B}}} – just as any subset of X {\displaystyle X} that includes a neighborhood of x {\displaystyle x} will necessarily be a neighborhood of x {\displaystyle x} (this follows from Int X B ⊆ Int X S {\displaystyle \operatorname {Int} _{X}B\subseteq \operatorname {Int} _{X}S} and the definition of "a neighborhood of x {\displaystyle x} "). Generalizing sequence convergence by using sets − determining sequence convergence without the sequence A sequence in X {\displaystyle X} is by definition a map N → X {\displaystyle \mathbb {N} \to X} from the natural numbers into the space X . {\displaystyle X.} The original notion of convergence in a topological space was that of a sequence converging to some given point in a space, such as a metric space. With metrizable spaces (or more generally first-countable spaces or Fréchet–Urysohn spaces), sequences usually suffices to characterize, or "describe", most topological properties, such as the closures of subsets or continuity of functions. But there are many spaces where sequences can not be used to describe even basic topological properties like closure or continuity. This failure of sequences was the motivation for defining notions such as nets
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"page_id": 47516955,
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and filters, which never fail to characterize topological properties. Nets directly generalize the notion of a sequence since nets are, by definition, maps I → X {\displaystyle I\to X} from an arbitrary directed set ( I , ≤ ) {\displaystyle (I,\leq )} into the space X . {\displaystyle X.} A sequence is just a net whose domain is I = N {\displaystyle I=\mathbb {N} } with the natural ordering. Nets have their own notion of convergence, which is a direct generalization of sequence convergence. Filters generalize sequence convergence in a different way by considering only the values of a sequence. To see how this is done, consider a sequence x ∙ = ( x i ) i = 1 ∞ in X , {\displaystyle x_{\bullet }=\left(x_{i}\right)_{i=1}^{\infty }{\text{ in }}X,} which is by definition just a function x ∙ : N → X {\displaystyle x_{\bullet }:\mathbb {N} \to X} whose value at i ∈ N {\displaystyle i\in \mathbb {N} } is denoted by x i {\displaystyle x_{i}} rather than by the usual parentheses notation x ∙ ( i ) {\displaystyle x_{\bullet }(i)} that is commonly used for arbitrary functions. Knowing only the image (sometimes called "the range") Im x ∙ := { x i : i ∈ N } = { x 1 , x 2 , … } {\displaystyle \operatorname {Im} x_{\bullet }:=\left\{x_{i}:i\in \mathbb {N} \right\}=\left\{x_{1},x_{2},\ldots \right\}} of the sequence is not enough to characterize its convergence; multiple sets are needed. It turns out that the needed sets are the following, which are called the tails of the sequence x ∙ {\displaystyle x_{\bullet }} : x ≥ 1 = { x 1 , x 2 , x 3 , x 4 , … } x ≥ 2 = { x 2 , x 3 , x 4 , x
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5 , … } x ≥ 3 = { x 3 , x 4 , x 5 , x 6 , … } ⋮ x ≥ n = { x n , x n + 1 , x n + 2 , x n + 3 , … } ⋮ {\displaystyle {\begin{alignedat}{8}x_{\geq 1}=\;&\{&&x_{1},&&x_{2},&&x_{3},&&x_{4},&&\ldots &&\,\}\\[0.3ex]x_{\geq 2}=\;&\{&&x_{2},&&x_{3},&&x_{4},&&x_{5},&&\ldots &&\,\}\\[0.3ex]x_{\geq 3}=\;&\{&&x_{3},&&x_{4},&&x_{5},&&x_{6},&&\ldots &&\,\}\\[0.3ex]&&&&&&&\;\,\vdots &&&&&&\\[0.3ex]x_{\geq n}=\;&\{&&x_{n},\;\;\,&&x_{n+1},\;&&x_{n+2},\;&&x_{n+3},&&\ldots &&\,\}\\[0.3ex]&&&&&&&\;\,\vdots &&&&&&\\[0.3ex]\end{alignedat}}} These sets completely determine this sequence's convergence (or non-convergence) because given any point, this sequence converges to it if and only if for every neighborhood U {\displaystyle U} (of this point), there is some integer n {\displaystyle n} such that U {\displaystyle U} contains all of the points x n , x n + 1 , … . {\displaystyle x_{n},x_{n+1},\ldots .} This can be reworded as: every neighborhood U {\displaystyle U} must contain some set of the form { x n , x n + 1 , … } {\displaystyle \{x_{n},x_{n+1},\ldots \}} as a subset. Or more briefly: every neighborhood must contain some tail x ≥ n {\displaystyle x_{\geq n}} as a subset. It is this characterization that can be used with the above family of tails to determine convergence (or non-convergence) of the sequence x ∙ : N → X . {\displaystyle x_{\bullet }:\mathbb {N} \to X.} Specifically, with the family of sets { x ≥ 1 , x ≥ 2 , … } {\displaystyle \{x_{\geq 1},x_{\geq 2},\ldots \}} in hand, the function x ∙ : N → X {\displaystyle x_{\bullet }:\mathbb {N} \to X} is no longer needed to determine convergence of this sequence (no matter what topology is placed on X {\displaystyle X} ). By generalizing this observation, the notion of "convergence" can be extended from sequences/functions to families of sets. The above set of tails of a sequence is in general not a filter
|
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"page_id": 47516955,
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but it does "generate" a filter via taking its upward closure (which consists of all supersets of all tails). The same is true of other important families of sets such as any neighborhood basis at a given point, which in general is also not a filter but does generate a filter via its upward closure (in particular, it generates the neighborhood filter at that point). The properties that these families share led to the notion of a filter base, also called a prefilter, which by definition is any family having the minimal properties necessary and sufficient for it to generate a filter via taking its upward closure. Nets versus filters − advantages and disadvantages Filters and nets each have their own advantages and drawbacks and there's no reason to use one notion exclusively over the other. Depending on what is being proved, a proof may be made significantly easier by using one of these notions instead of the other. Both filters and nets can be used to completely characterize any given topology. Nets are direct generalizations of sequences and can often be used similarly to sequences, so the learning curve for nets is typically much less steep than that for filters. However, filters, and especially ultrafilters, have many more uses outside of topology, such as in set theory, mathematical logic, model theory (ultraproducts, for example), abstract algebra, combinatorics, dynamics, order theory, generalized convergence spaces, Cauchy spaces, and in the definition and use of hyperreal numbers. Like sequences, nets are functions and so they have the advantages of functions. For example, like sequences, nets can be "plugged into" other functions, where "plugging in" is just function composition. Theorems related to functions and function composition may then be applied to nets. One example is the universal property of inverse limits, which is
|
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"page_id": 47516955,
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defined in terms of composition of functions rather than sets and it is more readily applied to functions like nets than to sets like filters (a prominent example of an inverse limit is the Cartesian product). Filters may be awkward to use in certain situations, such as when switching between a filter on a space X {\displaystyle X} and a filter on a dense subspace S ⊆ X . {\displaystyle S\subseteq X.} In contrast to nets, filters (and prefilters) are families of sets and so they have the advantages of sets. For example, if f {\displaystyle f} is surjective then the image f − 1 ( B ) := { f − 1 ( B ) : B ∈ B } {\displaystyle f^{-1}({\mathcal {B}}):=\left\{f^{-1}(B)~:~B\in {\mathcal {B}}\right\}} under f − 1 {\displaystyle f^{-1}} of an arbitrary filter or prefilter B {\displaystyle {\mathcal {B}}} is both easily defined and guaranteed to be a prefilter on f {\displaystyle f} 's domain, whereas it is less clear how to pullback (unambiguously/without choice) an arbitrary sequence (or net) y ∙ {\displaystyle y_{\bullet }} so as to obtain a sequence or net in the domain (unless f {\displaystyle f} is also injective and consequently a bijection, which is a stringent requirement). Similarly, the intersection of any collection of filters is once again a filter whereas it is not clear what this could mean for sequences or nets. Because filters are composed of subsets of the very topological space X {\displaystyle X} that is under consideration, topological set operations (such as closure or interior) may be applied to the sets that constitute the filter. Taking the closure of all the sets in a filter is sometimes useful in functional analysis for instance. Theorems and results about images or preimages of sets under a function may also be
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"page_id": 47516955,
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applied to the sets that constitute a filter; an example of such a result might be one of continuity's characterizations in terms of preimages of open/closed sets or in terms of the interior/closure operators. Special types of filters called ultrafilters have many useful properties that can significantly help in proving results. One downside of nets is their dependence on the directed sets that constitute their domains, which in general may be entirely unrelated to the space X . {\displaystyle X.} In fact, the class of nets in a given set X {\displaystyle X} is too large to even be a set (it is a proper class); this is because nets in X {\displaystyle X} can have domains of any cardinality. In contrast, the collection of all filters (and of all prefilters) on X {\displaystyle X} is a set whose cardinality is no larger than that of ℘ ( ℘ ( X ) ) . {\displaystyle \wp (\wp (X)).} Similar to a topology on X , {\displaystyle X,} a filter on X {\displaystyle X} is "intrinsic to X {\displaystyle X} " in the sense that both structures consist entirely of subsets of X {\displaystyle X} and neither definition requires any set that cannot be constructed from X {\displaystyle X} (such as N {\displaystyle \mathbb {N} } or other directed sets, which sequences and nets require). == Preliminaries, notation, and basic notions == In this article, upper case Roman letters like S {\displaystyle S} and X {\displaystyle X} denote sets (but not families unless indicated otherwise) and ℘ ( X ) {\displaystyle \wp (X)} will denote the power set of X . {\displaystyle X.} A subset of a power set is called a family of sets (or simply, a family) where it is over X {\displaystyle X} if it is a subset
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"page_id": 47516955,
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of ℘ ( X ) . {\displaystyle \wp (X).} Families of sets will be denoted by upper case calligraphy letters such as B {\displaystyle {\mathcal {B}}} , C {\displaystyle {\mathcal {C}}} , and F {\displaystyle {\mathcal {F}}} . Whenever these assumptions are needed, then it should be assumed that X {\displaystyle X} is non-empty and that B , F , {\displaystyle {\mathcal {B}},{\mathcal {F}},} etc. are families of sets over X . {\displaystyle X.} The terms "prefilter" and "filter base" are synonyms and will be used interchangeably. Warning about competing definitions and notation There are unfortunately several terms in the theory of filters that are defined differently by different authors. These include some of the most important terms such as "filter." While different definitions of the same term usually have significant overlap, due to the very technical nature of filters (and point–set topology), these differences in definitions nevertheless often have important consequences. When reading mathematical literature, it is recommended that readers check how the terminology related to filters is defined by the author. For this reason, this article will clearly state all definitions as they are used. Unfortunately, not all notation related to filters is well established and some notation varies greatly across the literature (for example, the notation for the set of all prefilters on a set) so in such cases this article uses whatever notation is most self describing or easily remembered. The theory of filters and prefilters is well developed and has a plethora of definitions and notations, many of which are now unceremoniously listed to prevent this article from becoming prolix and to allow for the easy look up of notation and definitions. Their important properties are described later. Sets operations The upward closure or isotonization in X {\displaystyle X} of a family of sets B
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⊆ ℘ ( X ) {\displaystyle {\mathcal {B}}\subseteq \wp (X)} is and similarly the downward closure of B {\displaystyle {\mathcal {B}}} is B ↓ := { S ⊆ B : B ∈ B } = ⋃ B ∈ B ℘ ( B ) . {\displaystyle {\mathcal {B}}^{\downarrow }:=\{S\subseteq B~:~B\in {\mathcal {B}}\,\}={\textstyle \bigcup \limits _{B\in {\mathcal {B}}}}\wp (B).} Throughout, f {\displaystyle f} is a map. Topology notation Denote the set of all topologies on a set X by Top ( X ) . {\displaystyle X{\text{ by }}\operatorname {Top} (X).} Suppose τ ∈ Top ( X ) , {\displaystyle \tau \in \operatorname {Top} (X),} S ⊆ X {\displaystyle S\subseteq X} is any subset, and x ∈ X {\displaystyle x\in X} is any point. If ∅ ≠ S ⊆ X {\displaystyle \varnothing \neq S\subseteq X} then τ ( S ) = ⋂ s ∈ S τ ( s ) and N τ ( S ) = ⋂ s ∈ S N τ ( s ) . {\displaystyle \tau (S)={\textstyle \bigcap \limits _{s\in S}}\tau (s){\text{ and }}{\mathcal {N}}_{\tau }(S)={\textstyle \bigcap \limits _{s\in S}}{\mathcal {N}}_{\tau }(s).} Nets and their tails A directed set is a set I {\displaystyle I} together with a preorder, which will be denoted by ≤ {\displaystyle \,\leq \,} (unless explicitly indicated otherwise), that makes ( I , ≤ ) {\displaystyle (I,\leq )} into an (upward) directed set; this means that for all i , j ∈ I , {\displaystyle i,j\in I,} there exists some k ∈ I {\displaystyle k\in I} such that i ≤ k and j ≤ k . {\displaystyle i\leq k{\text{ and }}j\leq k.} For any indices i and j , {\displaystyle i{\text{ and }}j,} the notation j ≥ i {\displaystyle j\geq i} is defined to mean i ≤ j {\displaystyle i\leq j} while i < j
|
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"page_id": 47516955,
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{\displaystyle i<j} is defined to mean that i ≤ j {\displaystyle i\leq j} holds but it is not true that j ≤ i {\displaystyle j\leq i} (if ≤ {\displaystyle \,\leq \,} is antisymmetric then this is equivalent to i ≤ j and i ≠ j {\displaystyle i\leq j{\text{ and }}i\neq j} ). A net in X {\displaystyle X} is a map from a non-empty directed set into X . {\displaystyle X.} The notation x ∙ = ( x i ) i ∈ I {\displaystyle x_{\bullet }=\left(x_{i}\right)_{i\in I}} will be used to denote a net with domain I . {\displaystyle I.} Warning about using strict comparison If x ∙ = ( x i ) i ∈ I {\displaystyle x_{\bullet }=\left(x_{i}\right)_{i\in I}} is a net and i ∈ I {\displaystyle i\in I} then it is possible for the set x > i = { x j : j > i and j ∈ I } , {\displaystyle x_{>i}=\left\{x_{j}~:~j>i{\text{ and }}j\in I\right\},} which is called the tail of x ∙ {\displaystyle x_{\bullet }} after i {\displaystyle i} , to be empty (for example, this happens if i {\displaystyle i} is an upper bound of the directed set I {\displaystyle I} ). In this case, the family { x > i : i ∈ I } {\displaystyle \left\{x_{>i}~:~i\in I\right\}} would contain the empty set, which would prevent it from being a prefilter (defined later). This is the (important) reason for defining Tails ( x ∙ ) {\displaystyle \operatorname {Tails} \left(x_{\bullet }\right)} as { x ≥ i : i ∈ I } {\displaystyle \left\{x_{\geq i}~:~i\in I\right\}} rather than { x > i : i ∈ I } {\displaystyle \left\{x_{>i}~:~i\in I\right\}} or even { x > i : i ∈ I } ∪ { x ≥ i : i ∈ I } {\displaystyle \left\{x_{>i}~:~i\in I\right\}\cup
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"page_id": 47516955,
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\left\{x_{\geq i}~:~i\in I\right\}} and it is for this reason that in general, when dealing with the prefilter of tails of a net, the strict inequality < {\displaystyle \,<\,} may not be used interchangeably with the inequality ≤ . {\displaystyle \,\leq .} === Filters and prefilters === The following is a list of properties that a family B {\displaystyle {\mathcal {B}}} of sets may possess and they form the defining properties of filters, prefilters, and filter subbases. Whenever it is necessary, it should be assumed that B ⊆ ℘ ( X ) . {\displaystyle {\mathcal {B}}\subseteq \wp (X).} Many of the properties of B {\displaystyle {\mathcal {B}}} defined above and below, such as "proper" and "directed downward," do not depend on X , {\displaystyle X,} so mentioning the set X {\displaystyle X} is optional when using such terms. Definitions involving being "upward closed in X , {\displaystyle X,} " such as that of "filter on X , {\displaystyle X,} " do depend on X {\displaystyle X} so the set X {\displaystyle X} should be mentioned if it is not clear from context. There are no prefilters on X = ∅ {\displaystyle X=\varnothing } (nor are there any nets valued in ∅ {\displaystyle \varnothing } ), which is why this article, like most authors, will automatically assume without comment that X ≠ ∅ {\displaystyle X\neq \varnothing } whenever this assumption is needed. ==== Basic examples ==== Named examples The singleton set B = { X } {\displaystyle {\mathcal {B}}=\{X\}} is called the indiscrete or trivial filter on X . {\displaystyle X.} It is the unique minimal filter on X {\displaystyle X} because it is a subset of every filter on X {\displaystyle X} ; however, it need not be a subset of every prefilter on X . {\displaystyle X.} The dual ideal
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℘ ( X ) {\displaystyle \wp (X)} is also called the degenerate filter on X {\displaystyle X} (despite not actually being a filter). It is the only dual ideal on X {\displaystyle X} that is not a filter on X . {\displaystyle X.} If ( X , τ ) {\displaystyle (X,\tau )} is a topological space and x ∈ X , {\displaystyle x\in X,} then the neighborhood filter N ( x ) {\displaystyle {\mathcal {N}}(x)} at x {\displaystyle x} is a filter on X . {\displaystyle X.} By definition, a family B ⊆ ℘ ( X ) {\displaystyle {\mathcal {B}}\subseteq \wp (X)} is called a neighborhood basis (resp. a neighborhood subbase) at x for ( X , τ ) {\displaystyle x{\text{ for }}(X,\tau )} if and only if B {\displaystyle {\mathcal {B}}} is a prefilter (resp. B {\displaystyle {\mathcal {B}}} is a filter subbase) and the filter on X {\displaystyle X} that B {\displaystyle {\mathcal {B}}} generates is equal to the neighborhood filter N ( x ) . {\displaystyle {\mathcal {N}}(x).} The subfamily τ ( x ) ⊆ N ( x ) {\displaystyle \tau (x)\subseteq {\mathcal {N}}(x)} of open neighborhoods is a filter base for N ( x ) . {\displaystyle {\mathcal {N}}(x).} Both prefilters N ( x ) and τ ( x ) {\displaystyle {\mathcal {N}}(x){\text{ and }}\tau (x)} also form a bases for topologies on X , {\displaystyle X,} with the topology generated τ ( x ) {\displaystyle \tau (x)} being coarser than τ . {\displaystyle \tau .} This example immediately generalizes from neighborhoods of points to neighborhoods of non-empty subsets S ⊆ X . {\displaystyle S\subseteq X.} B {\displaystyle {\mathcal {B}}} is an elementary prefilter if B = Tails ( x ∙ ) {\displaystyle {\mathcal {B}}=\operatorname {Tails} \left(x_{\bullet }\right)} for some sequence of points x ∙
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= ( x i ) i = 1 ∞ . {\displaystyle x_{\bullet }=\left(x_{i}\right)_{i=1}^{\infty }.} B {\displaystyle {\mathcal {B}}} is an elementary filter or a sequential filter on X {\displaystyle X} if B {\displaystyle {\mathcal {B}}} is a filter on X {\displaystyle X} generated by some elementary prefilter. The filter of tails generated by a sequence that is not eventually constant is necessarily not an ultrafilter. Every principal filter on a countable set is sequential as is every cofinite filter on a countably infinite set. The intersection of finitely many sequential filters is again sequential. The set F {\displaystyle {\mathcal {F}}} of all cofinite subsets of X {\displaystyle X} (meaning those sets whose complement in X {\displaystyle X} is finite) is proper if and only if F {\displaystyle {\mathcal {F}}} is infinite (or equivalently, X {\displaystyle X} is infinite), in which case F {\displaystyle {\mathcal {F}}} is a filter on X {\displaystyle X} known as the Fréchet filter or the cofinite filter on X . {\displaystyle X.} If X {\displaystyle X} is finite then F {\displaystyle {\mathcal {F}}} is equal to the dual ideal ℘ ( X ) , {\displaystyle \wp (X),} which is not a filter. If X {\displaystyle X} is infinite then the family { X ∖ { x } : x ∈ X } {\displaystyle \{X\setminus \{x\}~:~x\in X\}} of complements of singleton sets is a filter subbase that generates the Fréchet filter on X . {\displaystyle X.} As with any family of sets over X {\displaystyle X} that contains { X ∖ { x } : x ∈ X } , {\displaystyle \{X\setminus \{x\}~:~x\in X\},} the kernel of the Fréchet filter on X {\displaystyle X} is the empty set: ker F = ∅ . {\displaystyle \ker {\mathcal {F}}=\varnothing .} The intersection of all elements in any non-empty
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family F ⊆ Filters ( X ) {\displaystyle \mathbb {F} \subseteq \operatorname {Filters} (X)} is itself a filter on X {\displaystyle X} called the infimum or greatest lower bound of F in Filters ( X ) , {\displaystyle \mathbb {F} {\text{ in }}\operatorname {Filters} (X),} which is why it may be denoted by ⋀ F ∈ F F . {\displaystyle {\textstyle \bigwedge \limits _{{\mathcal {F}}\in \mathbb {F} }}{\mathcal {F}}.} Said differently, ker F = ⋂ F ∈ F F ∈ Filters ( X ) . {\displaystyle \ker \mathbb {F} ={\textstyle \bigcap \limits _{{\mathcal {F}}\in \mathbb {F} }}{\mathcal {F}}\in \operatorname {Filters} (X).} Because every filter on X {\displaystyle X} has { X } {\displaystyle \{X\}} as a subset, this intersection is never empty. By definition, the infimum is the finest/largest (relative to ⊆ and ≤ {\displaystyle \,\subseteq \,{\text{ and }}\,\leq \,} ) filter contained as a subset of each member of F . {\displaystyle \mathbb {F} .} If B and F {\displaystyle {\mathcal {B}}{\text{ and }}{\mathcal {F}}} are filters then their infimum in Filters ( X ) {\displaystyle \operatorname {Filters} (X)} is the filter B ( ∪ ) F . {\displaystyle {\mathcal {B}}\,(\cup )\,{\mathcal {F}}.} If B and F {\displaystyle {\mathcal {B}}{\text{ and }}{\mathcal {F}}} are prefilters then B ( ∪ ) F {\displaystyle {\mathcal {B}}\,(\cup )\,{\mathcal {F}}} is a prefilter that is coarser than both B and F {\displaystyle {\mathcal {B}}{\text{ and }}{\mathcal {F}}} (that is, B ( ∪ ) F ≤ B and B ( ∪ ) F ≤ F {\displaystyle {\mathcal {B}}\,(\cup )\,{\mathcal {F}}\leq {\mathcal {B}}{\text{ and }}{\mathcal {B}}\,(\cup )\,{\mathcal {F}}\leq {\mathcal {F}}} ); indeed, it is one of the finest such prefilters, meaning that if S {\displaystyle {\mathcal {S}}} is a prefilter such that S ≤ B and S ≤ F {\displaystyle
|
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"page_id": 47516955,
"source": null,
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{\mathcal {S}}\leq {\mathcal {B}}{\text{ and }}{\mathcal {S}}\leq {\mathcal {F}}} then necessarily S ≤ B ( ∪ ) F . {\displaystyle {\mathcal {S}}\leq {\mathcal {B}}\,(\cup )\,{\mathcal {F}}.} More generally, if B and F {\displaystyle {\mathcal {B}}{\text{ and }}{\mathcal {F}}} are non−empty families and if S := { S ⊆ ℘ ( X ) : S ≤ B and S ≤ F } {\displaystyle \mathbb {S} :=\{{\mathcal {S}}\subseteq \wp (X)~:~{\mathcal {S}}\leq {\mathcal {B}}{\text{ and }}{\mathcal {S}}\leq {\mathcal {F}}\}} then B ( ∪ ) F ∈ S {\displaystyle {\mathcal {B}}\,(\cup )\,{\mathcal {F}}\in \mathbb {S} } and B ( ∪ ) F {\displaystyle {\mathcal {B}}\,(\cup )\,{\mathcal {F}}} is a greatest element of ( S , ≤ ) . {\displaystyle (\mathbb {S} ,\leq ).} Let ∅ ≠ F ⊆ DualIdeals ( X ) {\displaystyle \varnothing \neq \mathbb {F} \subseteq \operatorname {DualIdeals} (X)} and let ∪ F = ⋃ F ∈ F F . {\displaystyle \cup \mathbb {F} ={\textstyle \bigcup \limits _{{\mathcal {F}}\in \mathbb {F} }}{\mathcal {F}}.} The supremum or least upper bound of F in DualIdeals ( X ) , {\displaystyle \mathbb {F} {\text{ in }}\operatorname {DualIdeals} (X),} denoted by ⋁ F ∈ F F , {\displaystyle {\textstyle \bigvee \limits _{{\mathcal {F}}\in \mathbb {F} }}{\mathcal {F}},} is the smallest (relative to ⊆ {\displaystyle \subseteq } ) dual ideal on X {\displaystyle X} containing every element of F {\displaystyle \mathbb {F} } as a subset; that is, it is the smallest (relative to ⊆ {\displaystyle \subseteq } ) dual ideal on X {\displaystyle X} containing ∪ F {\displaystyle \cup \mathbb {F} } as a subset. This dual ideal is ⋁ F ∈ F F = π ( ∪ F ) ↑ X , {\displaystyle {\textstyle \bigvee \limits _{{\mathcal {F}}\in \mathbb {F} }}{\mathcal {F}}=\pi \left(\cup \mathbb {F} \right)^{\uparrow X},} where π ( ∪ F )
|
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:= { F 1 ∩ ⋯ ∩ F n : n ∈ N and every F i belongs to some F ∈ F } {\displaystyle \pi \left(\cup \mathbb {F} \right):=\left\{F_{1}\cap \cdots \cap F_{n}~:~n\in \mathbb {N} {\text{ and every }}F_{i}{\text{ belongs to some }}{\mathcal {F}}\in \mathbb {F} \right\}} is the π-system generated by ∪ F . {\displaystyle \cup \mathbb {F} .} As with any non-empty family of sets, ∪ F {\displaystyle \cup \mathbb {F} } is contained in some filter on X {\displaystyle X} if and only if it is a filter subbase, or equivalently, if and only if ⋁ F ∈ F F = π ( ∪ F ) ↑ X {\displaystyle {\textstyle \bigvee \limits _{{\mathcal {F}}\in \mathbb {F} }}{\mathcal {F}}=\pi \left(\cup \mathbb {F} \right)^{\uparrow X}} is a filter on X , {\displaystyle X,} in which case this family is the smallest (relative to ⊆ {\displaystyle \subseteq } ) filter on X {\displaystyle X} containing every element of F {\displaystyle \mathbb {F} } as a subset and necessarily F ⊆ Filters ( X ) . {\displaystyle \mathbb {F} \subseteq \operatorname {Filters} (X).} Let ∅ ≠ F ⊆ Filters ( X ) {\displaystyle \varnothing \neq \mathbb {F} \subseteq \operatorname {Filters} (X)} and let ∪ F = ⋃ F ∈ F F . {\displaystyle \cup \mathbb {F} ={\textstyle \bigcup \limits _{{\mathcal {F}}\in \mathbb {F} }}{\mathcal {F}}.} The supremum or least upper bound of F in Filters ( X ) , {\displaystyle \mathbb {F} {\text{ in }}\operatorname {Filters} (X),} denoted by ⋁ F ∈ F F {\displaystyle {\textstyle \bigvee \limits _{{\mathcal {F}}\in \mathbb {F} }}{\mathcal {F}}} if it exists, is by definition the smallest (relative to ⊆ {\displaystyle \subseteq } ) filter on X {\displaystyle X} containing every element of F {\displaystyle \mathbb {F} } as a subset. If it
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exists then necessarily ⋁ F ∈ F F = π ( ∪ F ) ↑ X {\displaystyle {\textstyle \bigvee \limits _{{\mathcal {F}}\in \mathbb {F} }}{\mathcal {F}}=\pi \left(\cup \mathbb {F} \right)^{\uparrow X}} (as defined above) and ⋁ F ∈ F F {\displaystyle {\textstyle \bigvee \limits _{{\mathcal {F}}\in \mathbb {F} }}{\mathcal {F}}} will also be equal to the intersection of all filters on X {\displaystyle X} containing ∪ F . {\displaystyle \cup \mathbb {F} .} This supremum of F in Filters ( X ) {\displaystyle \mathbb {F} {\text{ in }}\operatorname {Filters} (X)} exists if and only if the dual ideal π ( ∪ F ) ↑ X {\displaystyle \pi \left(\cup \mathbb {F} \right)^{\uparrow X}} is a filter on X . {\displaystyle X.} The least upper bound of a family of filters F {\displaystyle \mathbb {F} } may fail to be a filter. Indeed, if X {\displaystyle X} contains at least two distinct elements then there exist filters B and C on X {\displaystyle {\mathcal {B}}{\text{ and }}{\mathcal {C}}{\text{ on }}X} for which there does not exist a filter F on X {\displaystyle {\mathcal {F}}{\text{ on }}X} that contains both B and C . {\displaystyle {\mathcal {B}}{\text{ and }}{\mathcal {C}}.} If ∪ F {\displaystyle \cup \mathbb {F} } is not a filter subbase then the supremum of F in Filters ( X ) {\displaystyle \mathbb {F} {\text{ in }}\operatorname {Filters} (X)} does not exist and the same is true of its supremum in Prefilters ( X ) {\displaystyle \operatorname {Prefilters} (X)} but their supremum in the set of all dual ideals on X {\displaystyle X} will exist (it being the degenerate filter ℘ ( X ) {\displaystyle \wp (X)} ). If B and F {\displaystyle {\mathcal {B}}{\text{ and }}{\mathcal {F}}} are prefilters (resp. filters on X {\displaystyle X} ) then
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B ( ∩ ) F {\displaystyle {\mathcal {B}}\,(\cap )\,{\mathcal {F}}} is a prefilter (resp. a filter) if and only if it is non-degenerate (or said differently, if and only if B and F {\displaystyle {\mathcal {B}}{\text{ and }}{\mathcal {F}}} mesh), in which case it is one of the coarsest prefilters (resp. the coarsest filter) on X {\displaystyle X} that is finer (with respect to ≤ {\displaystyle \,\leq } ) than both B and F ; {\displaystyle {\mathcal {B}}{\text{ and }}{\mathcal {F}};} this means that if S {\displaystyle {\mathcal {S}}} is any prefilter (resp. any filter) such that B ≤ S and F ≤ S {\displaystyle {\mathcal {B}}\leq {\mathcal {S}}{\text{ and }}{\mathcal {F}}\leq {\mathcal {S}}} then necessarily B ( ∩ ) F ≤ S , {\displaystyle {\mathcal {B}}\,(\cap )\,{\mathcal {F}}\leq {\mathcal {S}},} in which case it is denoted by B ∨ F . {\displaystyle {\mathcal {B}}\vee {\mathcal {F}}.} Other examples Let X = { p , 1 , 2 , 3 } {\displaystyle X=\{p,1,2,3\}} and let B = { { p } , { p , 1 , 2 } , { p , 1 , 3 } } , {\displaystyle {\mathcal {B}}=\{\{p\},\{p,1,2\},\{p,1,3\}\},} which makes B {\displaystyle {\mathcal {B}}} a prefilter and a filter subbase that is not closed under finite intersections. Because B {\displaystyle {\mathcal {B}}} is a prefilter, the smallest prefilter containing B {\displaystyle {\mathcal {B}}} is B . {\displaystyle {\mathcal {B}}.} The π-system generated by B {\displaystyle {\mathcal {B}}} is { { p , 1 } } ∪ B . {\displaystyle \{\{p,1\}\}\cup {\mathcal {B}}.} In particular, the smallest prefilter containing the filter subbase B {\displaystyle {\mathcal {B}}} is not equal to the set of all finite intersections of sets in B . {\displaystyle {\mathcal {B}}.} The filter on X {\displaystyle X} generated by B {\displaystyle {\mathcal {B}}} is
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B ↑ X = { S ⊆ X : p ∈ S } = { { p } ∪ T : T ⊆ { 1 , 2 , 3 } } . {\displaystyle {\mathcal {B}}^{\uparrow X}=\{S\subseteq X:p\in S\}=\{\{p\}\cup T~:~T\subseteq \{1,2,3\}\}.} All three of B , {\displaystyle {\mathcal {B}},} the π-system B {\displaystyle {\mathcal {B}}} generates, and B ↑ X {\displaystyle {\mathcal {B}}^{\uparrow X}} are examples of fixed, principal, ultra prefilters that are principal at the point p ; B ↑ X {\displaystyle p;{\mathcal {B}}^{\uparrow X}} is also an ultrafilter on X . {\displaystyle X.} Let ( X , τ ) {\displaystyle (X,\tau )} be a topological space, B ⊆ ℘ ( X ) , {\displaystyle {\mathcal {B}}\subseteq \wp (X),} and define B ¯ := { cl X B : B ∈ B } , {\displaystyle {\overline {\mathcal {B}}}:=\left\{\operatorname {cl} _{X}B~:~B\in {\mathcal {B}}\right\},} where B {\displaystyle {\mathcal {B}}} is necessarily finer than B ¯ . {\displaystyle {\overline {\mathcal {B}}}.} If B {\displaystyle {\mathcal {B}}} is non-empty (resp. non-degenerate, a filter subbase, a prefilter, closed under finite unions) then the same is true of B ¯ . {\displaystyle {\overline {\mathcal {B}}}.} If B {\displaystyle {\mathcal {B}}} is a filter on X {\displaystyle X} then B ¯ {\displaystyle {\overline {\mathcal {B}}}} is a prefilter but not necessarily a filter on X {\displaystyle X} although ( B ¯ ) ↑ X {\displaystyle \left({\overline {\mathcal {B}}}\right)^{\uparrow X}} is a filter on X {\displaystyle X} equivalent to B ¯ . {\displaystyle {\overline {\mathcal {B}}}.} The set B {\displaystyle {\mathcal {B}}} of all dense open subsets of a (non-empty) topological space X {\displaystyle X} is a proper π-system and so also a prefilter. If the space is a Baire space, then the set of all countable intersections of dense open subsets is a π-system and a
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prefilter that is finer than B . {\displaystyle {\mathcal {B}}.} If X = R n {\displaystyle X=\mathbb {R} ^{n}} (with 1 ≤ n ∈ N {\displaystyle 1\leq n\in \mathbb {N} } ) then the set B LebFinite {\displaystyle {\mathcal {B}}_{\operatorname {LebFinite} }} of all B ∈ B {\displaystyle B\in {\mathcal {B}}} such that B {\displaystyle B} has finite Lebesgue measure is a proper π-system and a free prefilter that is also a proper subset of B . {\displaystyle {\mathcal {B}}.} The prefilters B LebFinite {\displaystyle {\mathcal {B}}_{\operatorname {LebFinite} }} and B {\displaystyle {\mathcal {B}}} are equivalent and so generate the same filter on X . {\displaystyle X.} Since X {\displaystyle X} is a Baire space, every countable intersection of sets in B LebFinite {\displaystyle {\mathcal {B}}_{\operatorname {LebFinite} }} is dense in X {\displaystyle X} (and also comeagre and non-meager) so the set of all countable intersections of elements of B LebFinite {\displaystyle {\mathcal {B}}_{\operatorname {LebFinite} }} is a prefilter and π-system; it is also finer than, and not equivalent to, B LebFinite . {\displaystyle {\mathcal {B}}_{\operatorname {LebFinite} }.} ==== Ultrafilters ==== There are many other characterizations of "ultrafilter" and "ultra prefilter," which are listed in the article on ultrafilters. Important properties of ultrafilters are also described in that article. The ultrafilter lemma The following important theorem is due to Alfred Tarski (1930). A consequence of the ultrafilter lemma is that every filter is equal to the intersection of all ultrafilters containing it. Assuming the axioms of Zermelo–Fraenkel (ZF), the ultrafilter lemma follows from the Axiom of choice (in particular from Zorn's lemma) but is strictly weaker than it. The ultrafilter lemma implies the Axiom of choice for finite sets. If only dealing with Hausdorff spaces, then most basic results (as encountered in introductory courses) in Topology (such as Tychonoff's theorem
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"page_id": 47516955,
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for compact Hausdorff spaces and the Alexander subbase theorem) and in functional analysis (such as the Hahn–Banach theorem) can be proven using only the ultrafilter lemma; the full strength of the axiom of choice might not be needed. ==== Kernels ==== The kernel is useful in classifying properties of prefilters and other families of sets. If B ⊆ ℘ ( X ) {\displaystyle {\mathcal {B}}\subseteq \wp (X)} then ker ( B ↑ X ) = ker B {\displaystyle \ker \left({\mathcal {B}}^{\uparrow X}\right)=\ker {\mathcal {B}}} and this set is also equal to the kernel of the π-system that is generated by B . {\displaystyle {\mathcal {B}}.} In particular, if B {\displaystyle {\mathcal {B}}} is a filter subbase then the kernels of all of the following sets are equal: (1) B , {\displaystyle {\mathcal {B}},} (2) the π-system generated by B , {\displaystyle {\mathcal {B}},} and (3) the filter generated by B . {\displaystyle {\mathcal {B}}.} If f {\displaystyle f} is a map then f ( ker B ) ⊆ ker f ( B ) and f − 1 ( ker B ) = ker f − 1 ( B ) . {\displaystyle f(\ker {\mathcal {B}})\subseteq \ker f({\mathcal {B}}){\text{ and }}f^{-1}(\ker {\mathcal {B}})=\ker f^{-1}({\mathcal {B}}).} Equivalent families have equal kernels. Two principal families are equivalent if and only if their kernels are equal. ===== Classifying families by their kernels ===== If B {\displaystyle {\mathcal {B}}} is a principal filter on X {\displaystyle X} then ∅ ≠ ker B ∈ B {\displaystyle \varnothing \neq \ker {\mathcal {B}}\in {\mathcal {B}}} and B = { ker B } ↑ X {\displaystyle {\mathcal {B}}=\{\ker {\mathcal {B}}\}^{\uparrow X}} and { ker B } {\displaystyle \{\ker {\mathcal {B}}\}} is also the smallest prefilter that generates B . {\displaystyle {\mathcal
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{B}}.} Family of examples: For any non-empty C ⊆ R , {\displaystyle C\subseteq \mathbb {R} ,} the family B C = { R ∖ ( r + C ) : r ∈ R } {\displaystyle {\mathcal {B}}_{C}=\{\mathbb {R} \setminus (r+C)~:~r\in \mathbb {R} \}} is free but it is a filter subbase if and only if no finite union of the form ( r 1 + C ) ∪ ⋯ ∪ ( r n + C ) {\displaystyle \left(r_{1}+C\right)\cup \cdots \cup \left(r_{n}+C\right)} covers R , {\displaystyle \mathbb {R} ,} in which case the filter that it generates will also be free. In particular, B C {\displaystyle {\mathcal {B}}_{C}} is a filter subbase if C {\displaystyle C} is countable (for example, C = Q , Z , {\displaystyle C=\mathbb {Q} ,\mathbb {Z} ,} the primes), a meager set in R , {\displaystyle \mathbb {R} ,} a set of finite measure, or a bounded subset of R . {\displaystyle \mathbb {R} .} If C {\displaystyle C} is a singleton set then B C {\displaystyle {\mathcal {B}}_{C}} is a subbase for the Fréchet filter on R . {\displaystyle \mathbb {R} .} ===== Characterizing fixed ultra prefilters ===== If a family of sets B {\displaystyle {\mathcal {B}}} is fixed (that is, ker B ≠ ∅ {\displaystyle \ker {\mathcal {B}}\neq \varnothing } ) then B {\displaystyle {\mathcal {B}}} is ultra if and only if some element of B {\displaystyle {\mathcal {B}}} is a singleton set, in which case B {\displaystyle {\mathcal {B}}} will necessarily be a prefilter. Every principal prefilter is fixed, so a principal prefilter B {\displaystyle {\mathcal {B}}} is ultra if and only if ker B {\displaystyle \ker {\mathcal {B}}} is a singleton set. Every filter on X {\displaystyle X} that is principal at a single point is an ultrafilter, and if
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in addition X {\displaystyle X} is finite, then there are no ultrafilters on X {\displaystyle X} other than these. The next theorem shows that every ultrafilter falls into one of two categories: either it is free or else it is a principal filter generated by a single point. === Finer/coarser, subordination, and meshing === The preorder ≤ {\displaystyle \,\leq \,} that is defined below is of fundamental importance for the use of prefilters (and filters) in topology. For instance, this preorder is used to define the prefilter equivalent of "subsequence", where " F ≥ C {\displaystyle {\mathcal {F}}\geq {\mathcal {C}}} " can be interpreted as " F {\displaystyle {\mathcal {F}}} is a subsequence of C {\displaystyle {\mathcal {C}}} " (so "subordinate to" is the prefilter equivalent of "subsequence of"). It is also used to define prefilter convergence in a topological space. The definition of B {\displaystyle {\mathcal {B}}} meshes with C , {\displaystyle {\mathcal {C}},} which is closely related to the preorder ≤ , {\displaystyle \,\leq ,} is used in topology to define cluster points. Two families of sets B and C {\displaystyle {\mathcal {B}}{\text{ and }}{\mathcal {C}}} mesh and are compatible, indicated by writing B # C , {\displaystyle {\mathcal {B}}\#{\mathcal {C}},} if B ∩ C ≠ ∅ for all B ∈ B and C ∈ C . {\displaystyle B\cap C\neq \varnothing {\text{ for all }}B\in {\mathcal {B}}{\text{ and }}C\in {\mathcal {C}}.} If B and C {\displaystyle {\mathcal {B}}{\text{ and }}{\mathcal {C}}} do not mesh then they are dissociated. If S ⊆ X and B ⊆ ℘ ( X ) {\displaystyle S\subseteq X{\text{ and }}{\mathcal {B}}\subseteq \wp (X)} then B and S {\displaystyle {\mathcal {B}}{\text{ and }}S} are said to mesh if B and { S } {\displaystyle {\mathcal {B}}{\text{ and }}\{S\}} mesh, or equivalently, if the trace of
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B on S , {\displaystyle {\mathcal {B}}{\text{ on }}S,} which is the family B | S = { B ∩ S : B ∈ B } , {\displaystyle {\mathcal {B}}{\big \vert }_{S}=\{B\cap S~:~B\in {\mathcal {B}}\},} does not contain the empty set, where the trace is also called the restriction of B to S . {\displaystyle {\mathcal {B}}{\text{ to }}S.} Example: If x i ∙ = ( x i n ) n = 1 ∞ {\displaystyle x_{i_{\bullet }}=\left(x_{i_{n}}\right)_{n=1}^{\infty }} is a subsequence of x ∙ = ( x i ) i = 1 ∞ {\displaystyle x_{\bullet }=\left(x_{i}\right)_{i=1}^{\infty }} then Tails ( x i ∙ ) {\displaystyle \operatorname {Tails} \left(x_{i_{\bullet }}\right)} is subordinate to Tails ( x ∙ ) ; {\displaystyle \operatorname {Tails} \left(x_{\bullet }\right);} in symbols: Tails ( x i ∙ ) ⊢ Tails ( x ∙ ) {\displaystyle \operatorname {Tails} \left(x_{i_{\bullet }}\right)\vdash \operatorname {Tails} \left(x_{\bullet }\right)} and also Tails ( x ∙ ) ≤ Tails ( x i ∙ ) . {\displaystyle \operatorname {Tails} \left(x_{\bullet }\right)\leq \operatorname {Tails} \left(x_{i_{\bullet }}\right).} Stated in plain English, the prefilter of tails of a subsequence is always subordinate to that of the original sequence. To see this, let C := x ≥ i ∈ Tails ( x ∙ ) {\displaystyle C:=x_{\geq i}\in \operatorname {Tails} \left(x_{\bullet }\right)} be arbitrary (or equivalently, let i ∈ N {\displaystyle i\in \mathbb {N} } be arbitrary) and it remains to show that this set contains some F := x i ≥ n ∈ Tails ( x i ∙ ) . {\displaystyle F:=x_{i_{\geq n}}\in \operatorname {Tails} \left(x_{i_{\bullet }}\right).} For the set x ≥ i = { x i , x i + 1 , … } {\displaystyle x_{\geq i}=\left\{x_{i},x_{i+1},\ldots \right\}} to contain x i ≥ n = { x i n ,
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x i n + 1 , … } , {\displaystyle x_{i_{\geq n}}=\left\{x_{i_{n}},x_{i_{n+1}},\ldots \right\},} it is sufficient to have i ≤ i n . {\displaystyle i\leq i_{n}.} Since i 1 < i 2 < ⋯ {\displaystyle i_{1}<i_{2}<\cdots } are strictly increasing integers, there exists n ∈ N {\displaystyle n\in \mathbb {N} } such that i n ≥ i , {\displaystyle i_{n}\geq i,} and so x ≥ i ⊇ x i ≥ n {\displaystyle x_{\geq i}\supseteq x_{i_{\geq n}}} holds, as desired. Consequently, TailsFilter ( x ∙ ) ⊆ TailsFilter ( x i ∙ ) . {\displaystyle \operatorname {TailsFilter} \left(x_{\bullet }\right)\subseteq \operatorname {TailsFilter} \left(x_{i_{\bullet }}\right).} The left hand side will be a strict/proper subset of the right hand side if (for instance) every point of x ∙ {\displaystyle x_{\bullet }} is unique (that is, when x ∙ : N → X {\displaystyle x_{\bullet }:\mathbb {N} \to X} is injective) and x i ∙ {\displaystyle x_{i_{\bullet }}} is the even-indexed subsequence ( x 2 , x 4 , x 6 , … ) {\displaystyle \left(x_{2},x_{4},x_{6},\ldots \right)} because under these conditions, every tail x i ≥ n = { x 2 n , x 2 n + 2 , x 2 n + 4 , … } {\displaystyle x_{i_{\geq n}}=\left\{x_{2n},x_{2n+2},x_{2n+4},\ldots \right\}} (for every n ∈ N {\displaystyle n\in \mathbb {N} } ) of the subsequence will belong to the right hand side filter but not to the left hand side filter. For another example, if B {\displaystyle {\mathcal {B}}} is any family then ∅ ≤ B ≤ B ≤ { ∅ } {\displaystyle \varnothing \leq {\mathcal {B}}\leq {\mathcal {B}}\leq \{\varnothing \}} always holds and furthermore, { ∅ } ≤ B if and only if ∅ ∈ B . {\displaystyle \{\varnothing \}\leq {\mathcal {B}}{\text{ if and only if }}\varnothing \in {\mathcal {B}}.} A non-empty
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family that is coarser than a filter subbase must itself be a filter subbase. Every filter subbase is coarser than both the π-system that it generates and the filter that it generates. If C and F {\displaystyle {\mathcal {C}}{\text{ and }}{\mathcal {F}}} are families such that C ≤ F , {\displaystyle {\mathcal {C}}\leq {\mathcal {F}},} the family C {\displaystyle {\mathcal {C}}} is ultra, and ∅ ∉ F , {\displaystyle \varnothing \not \in {\mathcal {F}},} then F {\displaystyle {\mathcal {F}}} is necessarily ultra. It follows that any family that is equivalent to an ultra family will necessarily be ultra. In particular, if C {\displaystyle {\mathcal {C}}} is a prefilter then either both C {\displaystyle {\mathcal {C}}} and the filter C ↑ X {\displaystyle {\mathcal {C}}^{\uparrow X}} it generates are ultra or neither one is ultra. The relation ≤ {\displaystyle \,\leq \,} is reflexive and transitive, which makes it into a preorder on ℘ ( ℘ ( X ) ) . {\displaystyle \wp (\wp (X)).} The relation ≤ on Filters ( X ) {\displaystyle \,\leq \,{\text{ on }}\operatorname {Filters} (X)} is antisymmetric but if X {\displaystyle X} has more than one point then it is not symmetric. ==== Equivalent families of sets ==== The preorder ≤ {\displaystyle \,\leq \,} induces its canonical equivalence relation on ℘ ( ℘ ( X ) ) , {\displaystyle \wp (\wp (X)),} where for all B , C ∈ ℘ ( ℘ ( X ) ) , {\displaystyle {\mathcal {B}},{\mathcal {C}}\in \wp (\wp (X)),} B {\displaystyle {\mathcal {B}}} is equivalent to C {\displaystyle {\mathcal {C}}} if any of the following equivalent conditions hold: C ≤ B and B ≤ C . {\displaystyle {\mathcal {C}}\leq {\mathcal {B}}{\text{ and }}{\mathcal {B}}\leq {\mathcal {C}}.} The upward closures of C and B {\displaystyle {\mathcal {C}}{\text{ and }}{\mathcal {B}}} are equal.
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Two upward closed (in X {\displaystyle X} ) subsets of ℘ ( X ) {\displaystyle \wp (X)} are equivalent if and only if they are equal. If B ⊆ ℘ ( X ) {\displaystyle {\mathcal {B}}\subseteq \wp (X)} then necessarily ∅ ≤ B ≤ ℘ ( X ) {\displaystyle \varnothing \leq {\mathcal {B}}\leq \wp (X)} and B {\displaystyle {\mathcal {B}}} is equivalent to B ↑ X . {\displaystyle {\mathcal {B}}^{\uparrow X}.} Every equivalence class other than { ∅ } {\displaystyle \{\varnothing \}} contains a unique representative (that is, element of the equivalence class) that is upward closed in X . {\displaystyle X.} Properties preserved between equivalent families Let B , C ∈ ℘ ( ℘ ( X ) ) {\displaystyle {\mathcal {B}},{\mathcal {C}}\in \wp (\wp (X))} be arbitrary and let F {\displaystyle {\mathcal {F}}} be any family of sets. If B and C {\displaystyle {\mathcal {B}}{\text{ and }}{\mathcal {C}}} are equivalent (which implies that ker B = ker C {\displaystyle \ker {\mathcal {B}}=\ker {\mathcal {C}}} ) then for each of the statements/properties listed below, either it is true of both B and C {\displaystyle {\mathcal {B}}{\text{ and }}{\mathcal {C}}} or else it is false of both B and C {\displaystyle {\mathcal {B}}{\text{ and }}{\mathcal {C}}} : Not empty Proper (that is, ∅ {\displaystyle \varnothing } is not an element) Moreover, any two degenerate families are necessarily equivalent. Filter subbase Prefilter In which case B and C {\displaystyle {\mathcal {B}}{\text{ and }}{\mathcal {C}}} generate the same filter on X {\displaystyle X} (that is, their upward closures in X {\displaystyle X} are equal). Free Principal Ultra Is equal to the trivial filter { X } {\displaystyle \{X\}} In words, this means that the only subset of ℘ ( X ) {\displaystyle \wp (X)} that is equivalent to the trivial filter
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is the trivial filter. In general, this conclusion of equality does not extend to non−trivial filters (one exception is when both families are filters). Meshes with F {\displaystyle {\mathcal {F}}} Is finer than F {\displaystyle {\mathcal {F}}} Is coarser than F {\displaystyle {\mathcal {F}}} Is equivalent to F {\displaystyle {\mathcal {F}}} Missing from the above list is the word "filter" because this property is not preserved by equivalence. However, if B and C {\displaystyle {\mathcal {B}}{\text{ and }}{\mathcal {C}}} are filters on X , {\displaystyle X,} then they are equivalent if and only if they are equal; this characterization does not extend to prefilters. Equivalence of prefilters and filter subbases If B {\displaystyle {\mathcal {B}}} is a prefilter on X {\displaystyle X} then the following families are always equivalent to each other: B {\displaystyle {\mathcal {B}}} ; the π-system generated by B {\displaystyle {\mathcal {B}}} ; the filter on X {\displaystyle X} generated by B {\displaystyle {\mathcal {B}}} ; and moreover, these three families all generate the same filter on X {\displaystyle X} (that is, the upward closures in X {\displaystyle X} of these families are equal). In particular, every prefilter is equivalent to the filter that it generates. By transitivity, two prefilters are equivalent if and only if they generate the same filter. Every prefilter is equivalent to exactly one filter on X , {\displaystyle X,} which is the filter that it generates (that is, the prefilter's upward closure). Said differently, every equivalence class of prefilters contains exactly one representative that is a filter. In this way, filters can be considered as just being distinguished elements of these equivalence classes of prefilters. A filter subbase that is not also a prefilter cannot be equivalent to the prefilter (or filter) that it generates. In contrast, every prefilter is equivalent to
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the filter that it generates. This is why prefilters can, by and large, be used interchangeably with the filters that they generate while filter subbases cannot. == Set theoretic properties and constructions relevant to topology == === Trace and meshing === If B {\displaystyle {\mathcal {B}}} is a prefilter (resp. filter) on X and S ⊆ X {\displaystyle X{\text{ and }}S\subseteq X} then the trace of B on S , {\displaystyle {\mathcal {B}}{\text{ on }}S,} which is the family B | S := B ( ∩ ) { S } , {\displaystyle {\mathcal {B}}{\big \vert }_{S}:={\mathcal {B}}(\cap )\{S\},} is a prefilter (resp. a filter) if and only if B and S {\displaystyle {\mathcal {B}}{\text{ and }}S} mesh (that is, ∅ ∉ B ( ∩ ) { S } {\displaystyle \varnothing \not \in {\mathcal {B}}(\cap )\{S\}} ), in which case the trace of B on S {\displaystyle {\mathcal {B}}{\text{ on }}S} is said to be induced by S {\displaystyle S} . The trace is always finer than the original family; that is, B ≤ B | S . {\displaystyle {\mathcal {B}}\leq {\mathcal {B}}{\big \vert }_{S}.} If B {\displaystyle {\mathcal {B}}} is ultra and if B and S {\displaystyle {\mathcal {B}}{\text{ and }}S} mesh then the trace B | S {\displaystyle {\mathcal {B}}{\big \vert }_{S}} is ultra. If B {\displaystyle {\mathcal {B}}} is an ultrafilter on X {\displaystyle X} then the trace of B on S {\displaystyle {\mathcal {B}}{\text{ on }}S} is a filter on S {\displaystyle S} if and only if S ∈ B . {\displaystyle S\in {\mathcal {B}}.} For example, suppose that B {\displaystyle {\mathcal {B}}} is a filter on X and S ⊆ X {\displaystyle X{\text{ and }}S\subseteq X} is such that S ≠ X and X ∖ S ∉ B . {\displaystyle S\neq X{\text{ and }}X\setminus S\not \in
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{\mathcal {B}}.} Then B and S {\displaystyle {\mathcal {B}}{\text{ and }}S} mesh and B ∪ { S } {\displaystyle {\mathcal {B}}\cup \{S\}} generates a filter on X {\displaystyle X} that is strictly finer than B . {\displaystyle {\mathcal {B}}.} When prefilters mesh Given non-empty families B and C , {\displaystyle {\mathcal {B}}{\text{ and }}{\mathcal {C}},} the family B ( ∩ ) C := { B ∩ C : B ∈ B and C ∈ C } {\displaystyle {\mathcal {B}}(\cap ){\mathcal {C}}:=\{B\cap C~:~B\in {\mathcal {B}}{\text{ and }}C\in {\mathcal {C}}\}} satisfies C ≤ B ( ∩ ) C {\displaystyle {\mathcal {C}}\leq {\mathcal {B}}(\cap ){\mathcal {C}}} and B ≤ B ( ∩ ) C . {\displaystyle {\mathcal {B}}\leq {\mathcal {B}}(\cap ){\mathcal {C}}.} If B ( ∩ ) C {\displaystyle {\mathcal {B}}(\cap ){\mathcal {C}}} is proper (resp. a prefilter, a filter subbase) then this is also true of both B and C . {\displaystyle {\mathcal {B}}{\text{ and }}{\mathcal {C}}.} In order to make any meaningful deductions about B ( ∩ ) C {\displaystyle {\mathcal {B}}(\cap ){\mathcal {C}}} from B and C , B ( ∩ ) C {\displaystyle {\mathcal {B}}{\text{ and }}{\mathcal {C}},{\mathcal {B}}(\cap ){\mathcal {C}}} needs to be proper (that is, ∅ ∉ B ( ∩ ) C , {\displaystyle \varnothing \not \in {\mathcal {B}}(\cap ){\mathcal {C}},} which is the motivation for the definition of "mesh". In this case, B ( ∩ ) C {\displaystyle {\mathcal {B}}(\cap ){\mathcal {C}}} is a prefilter (resp. filter subbase) if and only if this is true of both B and C . {\displaystyle {\mathcal {B}}{\text{ and }}{\mathcal {C}}.} Said differently, if B and C {\displaystyle {\mathcal {B}}{\text{ and }}{\mathcal {C}}} are prefilters then they mesh if and only if B ( ∩ ) C {\displaystyle {\mathcal {B}}(\cap ){\mathcal {C}}} is a prefilter. Generalizing gives a well known
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characterization of "mesh" entirely in terms of subordination (that is, ≤ {\displaystyle \,\leq \,} ): Two prefilters (resp. filter subbases) B and C {\displaystyle {\mathcal {B}}{\text{ and }}{\mathcal {C}}} mesh if and only if there exists a prefilter (resp. filter subbase) F {\displaystyle {\mathcal {F}}} such that C ≤ F {\displaystyle {\mathcal {C}}\leq {\mathcal {F}}} and B ≤ F . {\displaystyle {\mathcal {B}}\leq {\mathcal {F}}.} If the least upper bound of two filters B and C {\displaystyle {\mathcal {B}}{\text{ and }}{\mathcal {C}}} exists in Filters ( X ) {\displaystyle \operatorname {Filters} (X)} then this least upper bound is equal to B ( ∩ ) C . {\displaystyle {\mathcal {B}}(\cap ){\mathcal {C}}.} === Images and preimages under functions === Throughout, f : X → Y and g : Y → Z {\displaystyle f:X\to Y{\text{ and }}g:Y\to Z} will be maps between non-empty sets. Images of prefilters Let B ⊆ ℘ ( Y ) . {\displaystyle {\mathcal {B}}\subseteq \wp (Y).} Many of the properties that B {\displaystyle {\mathcal {B}}} may have are preserved under images of maps; notable exceptions include being upward closed, being closed under finite intersections, and being a filter, which are not necessarily preserved. Explicitly, if one of the following properties is true of B on Y , {\displaystyle {\mathcal {B}}{\text{ on }}Y,} then it will necessarily also be true of g ( B ) on g ( Y ) {\displaystyle g({\mathcal {B}}){\text{ on }}g(Y)} (although possibly not on the codomain Z {\displaystyle Z} unless g {\displaystyle g} is surjective): ultra, ultrafilter, filter, prefilter, filter subbase, dual ideal, upward closed, proper/non-degenerate, ideal, closed under finite unions, downward closed, directed upward. Moreover, if B ⊆ ℘ ( Y ) {\displaystyle {\mathcal {B}}\subseteq \wp (Y)} is a prefilter then so are both g ( B ) and g − 1
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( g ( B ) ) . {\displaystyle g({\mathcal {B}}){\text{ and }}g^{-1}(g({\mathcal {B}})).} The image under a map f : X → Y {\displaystyle f:X\to Y} of an ultra set B ⊆ ℘ ( X ) {\displaystyle {\mathcal {B}}\subseteq \wp (X)} is again ultra and if B {\displaystyle {\mathcal {B}}} is an ultra prefilter then so is f ( B ) . {\displaystyle f({\mathcal {B}}).} If B {\displaystyle {\mathcal {B}}} is a filter then g ( B ) {\displaystyle g({\mathcal {B}})} is a filter on the range g ( Y ) , {\displaystyle g(Y),} but it is a filter on the codomain Z {\displaystyle Z} if and only if g {\displaystyle g} is surjective. Otherwise it is just a prefilter on Z {\displaystyle Z} and its upward closure must be taken in Z {\displaystyle Z} to obtain a filter. The upward closure of g ( B ) in Z {\displaystyle g({\mathcal {B}}){\text{ in }}Z} is g ( B ) ↑ Z = { S ⊆ Z : B ⊆ g − 1 ( S ) for some B ∈ B } {\displaystyle g({\mathcal {B}})^{\uparrow Z}=\left\{S\subseteq Z~:~B\subseteq g^{-1}(S){\text{ for some }}B\in {\mathcal {B}}\right\}} where if B {\displaystyle {\mathcal {B}}} is upward closed in Y {\displaystyle Y} (that is, a filter) then this simplifies to: g ( B ) ↑ Z = { S ⊆ Z : g − 1 ( S ) ∈ B } . {\displaystyle g({\mathcal {B}})^{\uparrow Z}=\left\{S\subseteq Z~:~g^{-1}(S)\in {\mathcal {B}}\right\}.} If X ⊆ Y {\displaystyle X\subseteq Y} then taking g {\displaystyle g} to be the inclusion map X → Y {\displaystyle X\to Y} shows that any prefilter (resp. ultra prefilter, filter subbase) on X {\displaystyle X} is also a prefilter (resp. ultra prefilter, filter subbase) on Y . {\displaystyle Y.} Preimages of prefilters Let B ⊆ ℘ (
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}
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Y ) . {\displaystyle {\mathcal {B}}\subseteq \wp (Y).} Under the assumption that f : X → Y {\displaystyle f:X\to Y} is surjective: f − 1 ( B ) {\displaystyle f^{-1}({\mathcal {B}})} is a prefilter (resp. filter subbase, π-system, closed under finite unions, proper) if and only if this is true of B . {\displaystyle {\mathcal {B}}.} However, if B {\displaystyle {\mathcal {B}}} is an ultrafilter on Y {\displaystyle Y} then even if f {\displaystyle f} is surjective (which would make f − 1 ( B ) {\displaystyle f^{-1}({\mathcal {B}})} a prefilter), it is nevertheless still possible for the prefilter f − 1 ( B ) {\displaystyle f^{-1}({\mathcal {B}})} to be neither ultra nor a filter on X . {\displaystyle X.} If f : X → Y {\displaystyle f:X\to Y} is not surjective then denote the trace of B on f ( X ) {\displaystyle {\mathcal {B}}{\text{ on }}f(X)} by B | f ( X ) , {\displaystyle {\mathcal {B}}{\big \vert }_{f(X)},} where in this case particular case the trace satisfies: B | f ( X ) = f ( f − 1 ( B ) ) {\displaystyle {\mathcal {B}}{\big \vert }_{f(X)}=f\left(f^{-1}({\mathcal {B}})\right)} and consequently also: f − 1 ( B ) = f − 1 ( B | f ( X ) ) . {\displaystyle f^{-1}({\mathcal {B}})=f^{-1}\left({\mathcal {B}}{\big \vert }_{f(X)}\right).} This last equality and the fact that the trace B | f ( X ) {\displaystyle {\mathcal {B}}{\big \vert }_{f(X)}} is a family of sets over f ( X ) {\displaystyle f(X)} means that to draw conclusions about f − 1 ( B ) , {\displaystyle f^{-1}({\mathcal {B}}),} the trace B | f ( X ) {\displaystyle {\mathcal {B}}{\big \vert }_{f(X)}} can be used in place of B {\displaystyle {\mathcal {B}}} and the surjection f : X → f (
|
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"page_id": 47516955,
"source": null,
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X ) {\displaystyle f:X\to f(X)} can be used in place of f : X → Y . {\displaystyle f:X\to Y.} For example: f − 1 ( B ) {\displaystyle f^{-1}({\mathcal {B}})} is a prefilter (resp. filter subbase, π-system, proper) if and only if this is true of B | f ( X ) . {\displaystyle {\mathcal {B}}{\big \vert }_{f(X)}.} In this way, the case where f {\displaystyle f} is not (necessarily) surjective can be reduced down to the case of a surjective function (which is a case that was described at the start of this subsection). Even if B {\displaystyle {\mathcal {B}}} is an ultrafilter on Y , {\displaystyle Y,} if f {\displaystyle f} is not surjective then it is nevertheless possible that ∅ ∈ B | f ( X ) , {\displaystyle \varnothing \in {\mathcal {B}}{\big \vert }_{f(X)},} which would make f − 1 ( B ) {\displaystyle f^{-1}({\mathcal {B}})} degenerate as well. The next characterization shows that degeneracy is the only obstacle. If B {\displaystyle {\mathcal {B}}} is a prefilter then the following are equivalent: f − 1 ( B ) {\displaystyle f^{-1}({\mathcal {B}})} is a prefilter; B | f ( X ) {\displaystyle {\mathcal {B}}{\big \vert }_{f(X)}} is a prefilter; ∅ ∉ B | f ( X ) {\displaystyle \varnothing \not \in {\mathcal {B}}{\big \vert }_{f(X)}} ; B {\displaystyle {\mathcal {B}}} meshes with f ( X ) {\displaystyle f(X)} and moreover, if f − 1 ( B ) {\displaystyle f^{-1}({\mathcal {B}})} is a prefilter then so is f ( f − 1 ( B ) ) . {\displaystyle f\left(f^{-1}({\mathcal {B}})\right).} If S ⊆ Y {\displaystyle S\subseteq Y} and if In : S → Y {\displaystyle \operatorname {In} :S\to Y} denotes the inclusion map then the trace of B on S {\displaystyle {\mathcal {B}}{\text{ on }}S} is equal
|
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to In − 1 ( B ) . {\displaystyle \operatorname {In} ^{-1}({\mathcal {B}}).} This observation allows the results in this subsection to be applied to investigating the trace on a set. ==== Subordination is preserved by images and preimages ==== The relation ≤ {\displaystyle \,\leq \,} is preserved under both images and preimages of families of sets. This means that for any families C and F , {\displaystyle {\mathcal {C}}{\text{ and }}{\mathcal {F}},} C ≤ F implies g ( C ) ≤ g ( F ) and f − 1 ( C ) ≤ f − 1 ( F ) . {\displaystyle {\mathcal {C}}\leq {\mathcal {F}}\quad {\text{ implies }}\quad g({\mathcal {C}})\leq g({\mathcal {F}})\quad {\text{ and }}\quad f^{-1}({\mathcal {C}})\leq f^{-1}({\mathcal {F}}).} Moreover, the following relations always hold for any family of sets C {\displaystyle {\mathcal {C}}} : C ≤ f ( f − 1 ( C ) ) {\displaystyle {\mathcal {C}}\leq f\left(f^{-1}({\mathcal {C}})\right)} where equality will hold if f {\displaystyle f} is surjective. Furthermore, f − 1 ( C ) = f − 1 ( f ( f − 1 ( C ) ) ) and g ( C ) = g ( g − 1 ( g ( C ) ) ) . {\displaystyle f^{-1}({\mathcal {C}})=f^{-1}\left(f\left(f^{-1}({\mathcal {C}})\right)\right)\quad {\text{ and }}\quad g({\mathcal {C}})=g\left(g^{-1}(g({\mathcal {C}}))\right).} If B ⊆ ℘ ( X ) and C ⊆ ℘ ( Y ) {\displaystyle {\mathcal {B}}\subseteq \wp (X){\text{ and }}{\mathcal {C}}\subseteq \wp (Y)} then f ( B ) ≤ C if and only if B ≤ f − 1 ( C ) {\displaystyle f({\mathcal {B}})\leq {\mathcal {C}}\quad {\text{ if and only if }}\quad {\mathcal {B}}\leq f^{-1}({\mathcal {C}})} and g − 1 ( g ( C ) ) ≤ C {\displaystyle g^{-1}(g({\mathcal {C}}))\leq {\mathcal {C}}} where equality will hold if g {\displaystyle g} is injective. ===
|
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"page_id": 47516955,
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Products of prefilters === Suppose X ∙ = ( X i ) i ∈ I {\displaystyle X_{\bullet }=\left(X_{i}\right)_{i\in I}} is a family of one or more non-empty sets, whose product will be denoted by ∏ X ∙ := ∏ i ∈ I X i , {\displaystyle {\textstyle \prod _{}}X_{\bullet }:={\textstyle \prod \limits _{i\in I}}X_{i},} and for every index i ∈ I , {\displaystyle i\in I,} let Pr X i : ∏ X ∙ → X i {\displaystyle \Pr {}_{X_{i}}:\prod X_{\bullet }\to X_{i}} denote the canonical projection. Let B ∙ := ( B i ) i ∈ I {\displaystyle {\mathcal {B}}_{\bullet }:=\left({\mathcal {B}}_{i}\right)_{i\in I}} be non−empty families, also indexed by I , {\displaystyle I,} such that B i ⊆ ℘ ( X i ) {\displaystyle {\mathcal {B}}_{i}\subseteq \wp \left(X_{i}\right)} for each i ∈ I . {\displaystyle i\in I.} The product of the families B ∙ {\displaystyle {\mathcal {B}}_{\bullet }} is defined identically to how the basic open subsets of the product topology are defined (had all of these B i {\displaystyle {\mathcal {B}}_{i}} been topologies). That is, both the notations ∏ B ∙ = ∏ i ∈ I B i {\displaystyle \prod _{}{\mathcal {B}}_{\bullet }=\prod _{i\in I}{\mathcal {B}}_{i}} denote the family of all cylinder subsets ∏ i ∈ I S i ⊆ ∏ X ∙ {\displaystyle {\textstyle \prod \limits _{i\in I}}S_{i}\subseteq {\textstyle \prod }X_{\bullet }} such that S i = X i {\displaystyle S_{i}=X_{i}} for all but finitely many i ∈ I {\displaystyle i\in I} and where S i ∈ B i {\displaystyle S_{i}\in {\mathcal {B}}_{i}} for any one of these finitely many exceptions (that is, for any i {\displaystyle i} such that S i ≠ X i , {\displaystyle S_{i}\neq X_{i},} necessarily S i ∈ B i {\displaystyle S_{i}\in {\mathcal {B}}_{i}} ). When every B i {\displaystyle {\mathcal {B}}_{i}} is
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{
"page_id": 47516955,
"source": null,
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|
a filter subbase then the family ⋃ i ∈ I Pr X i − 1 ( B i ) {\displaystyle {\textstyle \bigcup \limits _{i\in I}}\Pr {}_{X_{i}}^{-1}\left({\mathcal {B}}_{i}\right)} is a filter subbase for the filter on ∏ X ∙ {\displaystyle {\textstyle \prod }X_{\bullet }} generated by B ∙ . {\displaystyle {\mathcal {B}}_{\bullet }.} If ∏ B ∙ {\displaystyle {\textstyle \prod }{\mathcal {B}}_{\bullet }} is a filter subbase then the filter on ∏ X ∙ {\displaystyle {\textstyle \prod }X_{\bullet }} that it generates is called the filter generated by B ∙ {\displaystyle {\mathcal {B}}_{\bullet }} . If every B i {\displaystyle {\mathcal {B}}_{i}} is a prefilter on X i {\displaystyle X_{i}} then ∏ B ∙ {\displaystyle {\textstyle \prod }{\mathcal {B}}_{\bullet }} will be a prefilter on ∏ X ∙ {\displaystyle {\textstyle \prod }X_{\bullet }} and moreover, this prefilter is equal to the coarsest prefilter F on ∏ X ∙ {\displaystyle {\mathcal {F}}{\text{ on }}{\textstyle \prod }X_{\bullet }} such that Pr X i ( F ) = B i {\displaystyle \Pr {}_{X_{i}}({\mathcal {F}})={\mathcal {B}}_{i}} for every i ∈ I . {\displaystyle i\in I.} However, ∏ B ∙ {\displaystyle {\textstyle \prod }{\mathcal {B}}_{\bullet }} may fail to be a filter on ∏ X ∙ {\displaystyle {\textstyle \prod }X_{\bullet }} even if every B i {\displaystyle {\mathcal {B}}_{i}} is a filter on X i . {\displaystyle X_{i}.} == Convergence, limits, and cluster points == Throughout, ( X , τ ) {\displaystyle (X,\tau )} is a topological space. Prefilters vs. filters With respect to maps and subsets, the property of being a prefilter is in general more well behaved and better preserved than the property of being a filter. For instance, the image of a prefilter under some map is again a prefilter; but the image of a filter under a non-surjective map is never a
|
{
"page_id": 47516955,
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filter on the codomain, although it will be a prefilter. The situation is the same with preimages under non-injective maps (even if the map is surjective). If S ⊆ X {\displaystyle S\subseteq X} is a proper subset then any filter on S {\displaystyle S} will not be a filter on X , {\displaystyle X,} although it will be a prefilter. One advantage that filters have is that they are distinguished representatives of their equivalence class (relative to ≤ {\displaystyle \,\leq } ), meaning that any equivalence class of prefilters contains a unique filter. This property may be useful when dealing with equivalence classes of prefilters (for instance, they are useful in the construction of completions of uniform spaces via Cauchy filters). The many properties that characterize ultrafilters are also often useful. They are used to, for example, construct the Stone–Čech compactification. The use of ultrafilters generally requires that the ultrafilter lemma be assumed. But in the many fields where the axiom of choice (or the Hahn–Banach theorem) is assumed, the ultrafilter lemma necessarily holds and does not require an addition assumption. A note on intuition Suppose that F {\displaystyle {\mathcal {F}}} is a non-principal filter on an infinite set X . {\displaystyle X.} F {\displaystyle {\mathcal {F}}} has one "upward" property (that of being closed upward) and one "downward" property (that of being directed downward). Starting with any F 0 ∈ F , {\displaystyle F_{0}\in {\mathcal {F}},} there always exists some F 1 ∈ F {\displaystyle F_{1}\in {\mathcal {F}}} that is a proper subset of F 0 {\displaystyle F_{0}} ; this may be continued ad infinitum to get a sequence F 0 ⊋ F 1 ⊋ ⋯ {\displaystyle F_{0}\supsetneq F_{1}\supsetneq \cdots } of sets in F {\displaystyle {\mathcal {F}}} with each F i + 1 {\displaystyle F_{i+1}} being a proper
|
{
"page_id": 47516955,
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subset of F i . {\displaystyle F_{i}.} The same is not true going "upward", for if F 0 = X ∈ F {\displaystyle F_{0}=X\in {\mathcal {F}}} then there is no set in F {\displaystyle {\mathcal {F}}} that contains X {\displaystyle X} as a proper subset. Thus when it comes to limiting behavior (which is a topic central to the field of topology), going "upward" leads to a dead end, while going "downward" is typically fruitful. So to gain understanding and intuition about how filters (and prefilter) relate to concepts in topology, the "downward" property is usually the one to concentrate on. This is also why so many topological properties can be described by using only prefilters, rather than requiring filters (which only differ from prefilters in that they are also upward closed). The "upward" property of filters is less important for topological intuition but it is sometimes useful to have for technical reasons. For example, with respect to ⊆ , {\displaystyle \,\subseteq ,} every filter subbase is contained in a unique smallest filter but there may not exist a unique smallest prefilter containing it. === Limits and convergence === A family B {\displaystyle {\mathcal {B}}} is said to converge in ( X , τ ) {\displaystyle (X,\tau )} to a point x {\displaystyle x} of X {\displaystyle X} if B ≥ N ( x ) . {\displaystyle {\mathcal {B}}\geq {\mathcal {N}}(x).} Explicitly, N ( x ) ≤ B {\displaystyle {\mathcal {N}}(x)\leq {\mathcal {B}}} means that every neighborhood N of x {\displaystyle N{\text{ of }}x} contains some B ∈ B {\displaystyle B\in {\mathcal {B}}} as a subset (that is, B ⊆ N {\displaystyle B\subseteq N} ); thus the following then holds: N ∋ N ⊇ B ∈ B . {\displaystyle {\mathcal {N}}\ni N\supseteq B\in {\mathcal {B}}.} In words, a family converges
|
{
"page_id": 47516955,
"source": null,
"title": "Filters in topology"
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to a point or subset x {\displaystyle x} if and only if it is finer than the neighborhood filter at x . {\displaystyle x.} A family B {\displaystyle {\mathcal {B}}} converging to a point x {\displaystyle x} may be indicated by writing B → x or lim B → x in X {\displaystyle {\mathcal {B}}\to x{\text{ or }}\lim {\mathcal {B}}\to x{\text{ in }}X} and saying that x {\displaystyle x} is a limit of B in X ; {\displaystyle {\mathcal {B}}{\text{ in }}X;} if this limit x {\displaystyle x} is a point (and not a subset), then x {\displaystyle x} is also called a limit point. As usual, lim B = x {\displaystyle \lim {\mathcal {B}}=x} is defined to mean that B → x {\displaystyle {\mathcal {B}}\to x} and x ∈ X {\displaystyle x\in X} is the only limit point of B ; {\displaystyle {\mathcal {B}};} that is, if also B → z then z = x . {\displaystyle {\mathcal {B}}\to z{\text{ then }}z=x.} (If the notation " lim B = x {\displaystyle \lim {\mathcal {B}}=x} " did not also require that the limit point x {\displaystyle x} be unique then the equals sign = would no longer be guaranteed to be transitive). The set of all limit points of B {\displaystyle {\mathcal {B}}} is denoted by lim X B or lim B . {\displaystyle \lim {}_{X}{\mathcal {B}}{\text{ or }}\lim {\mathcal {B}}.} In the above definitions, it suffices to check that B {\displaystyle {\mathcal {B}}} is finer than some (or equivalently, finer than every) neighborhood base in ( X , τ ) {\displaystyle (X,\tau )} of the point (for example, such as τ ( x ) = { U ∈ τ : x ∈ U } {\displaystyle \tau (x)=\{U\in \tau :x\in U\}} or τ ( S ) = ⋂ s ∈
|
{
"page_id": 47516955,
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S τ ( s ) {\displaystyle \tau (S)={\textstyle \bigcap \limits _{s\in S}}\tau (s)} when S ≠ ∅ {\displaystyle S\neq \varnothing } ). Examples If X := R n {\displaystyle X:=\mathbb {R} ^{n}} is Euclidean space and ‖ x ‖ {\displaystyle \|x\|} denotes the Euclidean norm (which is the distance from the origin, defined as usual), then all of the following families converge to the origin: the prefilter { B r ( 0 ) : 0 < r ≤ 1 } {\displaystyle \{B_{r}(0):0<r\leq 1\}} of all open balls centered at the origin, where B r ( z ) = { x : ‖ x − z ‖ < r } . {\displaystyle B_{r}(z)=\{x:\|x-z\|<r\}.} the prefilter { B ≤ r ( 0 ) : 0 < r ≤ 1 } {\displaystyle \{B_{\leq r}(0):0<r\leq 1\}} of all closed balls centered at the origin, where B ≤ r ( z ) = { x : ‖ x − z ‖ ≤ r } . {\displaystyle B_{\leq r}(z)=\{x:\|x-z\|\leq r\}.} This prefilter is equivalent to the one above. the prefilter { R ∩ B ≤ r ( 0 ) : 0 < r ≤ 1 } {\displaystyle \{R\cap B_{\leq r}(0):0<r\leq 1\}} where R = S 1 ∪ S 1 / 2 ∪ S 1 / 3 ∪ ⋯ {\displaystyle R=S_{1}\cup S_{1/2}\cup S_{1/3}\cup \cdots } is a union of spheres S r = { x : ‖ x ‖ = r } {\displaystyle S_{r}=\{x:\|x\|=r\}} centered at the origin having progressively smaller radii. This family consists of the sets S 1 / n ∪ S 1 / ( n + 1 ) ∪ S 1 / ( n + 2 ) ∪ ⋯ {\displaystyle S_{1/n}\cup S_{1/(n+1)}\cup S_{1/(n+2)}\cup \cdots } as n {\displaystyle n} ranges over the positive integers. any of the families above but with the radius r
|
{
"page_id": 47516955,
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{\displaystyle r} ranging over 1 , 1 / 2 , 1 / 3 , 1 / 4 , … {\displaystyle 1,\,1/2,\,1/3,\,1/4,\ldots } (or over any other positive decreasing sequence) instead of over all positive reals. Drawing or imagining any one of these sequences of sets when X = R 2 {\displaystyle X=\mathbb {R} ^{2}} has dimension n = 2 {\displaystyle n=2} suggests that intuitively, these sets "should" converge to the origin (and indeed they do). This is the intuition that the above definition of a "convergent prefilter" make rigorous. Although ‖ ⋅ ‖ {\displaystyle \|\cdot \|} was assumed to be the Euclidean norm, the example above remains valid for any other norm on R n . {\displaystyle \mathbb {R} ^{n}.} The one and only limit point in X := R {\displaystyle X:=\mathbb {R} } of the free prefilter { ( 0 , r ) : r > 0 } {\displaystyle \{(0,r):r>0\}} is 0 {\displaystyle 0} since every open ball around the origin contains some open interval of this form. The fixed prefilter B := { [ 0 , 1 + r ) : r > 0 } {\displaystyle {\mathcal {B}}:=\{[0,1+r):r>0\}} does not converges in R {\displaystyle \mathbb {R} } to any point and so lim B = ∅ , {\displaystyle \lim {\mathcal {B}}=\varnothing ,} although B {\displaystyle {\mathcal {B}}} does converge to the set ker B = [ 0 , 1 ] {\displaystyle \ker {\mathcal {B}}=[0,1]} since N ( [ 0 , 1 ] ) ≤ B . {\displaystyle {\mathcal {N}}([0,1])\leq {\mathcal {B}}.} However, not every fixed prefilter converges to its kernel. For instance, the fixed prefilter { [ 0 , 1 + r ) ∪ ( 1 + 1 / r , ∞ ) : r > 0 } {\displaystyle \{[0,1+r)\cup (1+1/r,\infty ):r>0\}} also has kernel [ 0
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{
"page_id": 47516955,
"source": null,
"title": "Filters in topology"
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