id stringlengths 11 15 | title stringlengths 7 171 | problem stringlengths 9 4.33k | solution stringlengths 6 19k | problem_wikitext stringlengths 9 4.42k | solution_wikitext stringlengths 7 19.1k | proof_title stringlengths 9 171 | theorem_url stringlengths 34 198 | proof_url stringlengths 36 198 | categories listlengths 0 9 | theorem_references listlengths 0 36 | proof_references listlengths 0 253 |
|---|---|---|---|---|---|---|---|---|---|---|---|
proofwiki-13800 | Condition for Repunits to be Coprime | Let $R_p$ and $R_q$ be repunit numbers with $p$ and $q$ digits respectively.
Then $R_p$ and $R_q$ are coprime {{iff}} $p$ and $q$ are coprime. | === Necessary Condition ===
We show the contrapositive.
Suppose $p$ and $q$ are not coprime.
Then there exists some $d > 1$ such that $p = d m$ and $q = d n$.
Then by Divisors of Repunit with Composite Index:
:$R_d \divides R_p$
and:
:$R_d \divides R_q$
where $\divides$ denotes divisibility.
That is, $R_d$ is a common ... | Let $R_p$ and $R_q$ be [[Definition:Repunit|repunit numbers]] with $p$ and $q$ [[Definition:Digit|digits]] respectively.
Then $R_p$ and $R_q$ are [[Definition:Coprime Integers|coprime]] {{iff}} $p$ and $q$ are [[Definition:Coprime Integers|coprime]]. | === Necessary Condition ===
We show the [[Definition:Contrapositive Statement|contrapositive]].
Suppose $p$ and $q$ are not [[Definition:Coprime Integers|coprime]].
Then there exists some $d > 1$ such that $p = d m$ and $q = d n$.
Then by [[Divisors of Repunit with Composite Index]]:
:$R_d \divides R_p$
and:
:$R_d ... | Condition for Repunits to be Coprime | https://proofwiki.org/wiki/Condition_for_Repunits_to_be_Coprime | https://proofwiki.org/wiki/Condition_for_Repunits_to_be_Coprime | [
"Repunits"
] | [
"Definition:Repunit",
"Definition:Digit",
"Definition:Coprime/Integers",
"Definition:Coprime/Integers"
] | [
"Definition:Contrapositive Statement",
"Definition:Coprime/Integers",
"Divisors of Repunit with Composite Index",
"Definition:Divisor (Algebra)/Integer",
"Definition:Common Divisor/Integers",
"Definition:Coprime/Integers",
"Definition:Contrapositive Statement",
"Definition:Coprime/Integers",
"Diviso... |
proofwiki-13801 | Divisors of Repunit with Composite Index | Let $R_n$ be a repunit number with $n$ digits.
Let $n$ be composite such that $n = r s$ where $1 < r < n$ and $1 < s < n$.
Then $R_r$ and $R_s$ are both divisors of $R_n$. | Let $n = r s$.
Then:
{{begin-eqn}}
{{eqn | l = R_n
| r = \sum_{k \mathop = 0}^{n - 1} 10^k
| c = Basis Representation Theorem
}}
{{eqn | r = \sum_{j \mathop = 0}^{s - 1} \paren {\sum_{k \mathop = 0}^{r - 1} 10^k} 10^{r j}
| c =
}}
{{eqn | r = \paren {\sum_{k \mathop = 0}^{r - 1} 10^k} \paren {\sum_{j... | Let $R_n$ be a [[Definition:Repunit|repunit number]] with $n$ [[Definition:Digit|digits]].
Let $n$ be [[Definition:Composite Number|composite]] such that $n = r s$ where $1 < r < n$ and $1 < s < n$.
Then $R_r$ and $R_s$ are both [[Definition:Divisor of Integer|divisors]] of $R_n$. | Let $n = r s$.
Then:
{{begin-eqn}}
{{eqn | l = R_n
| r = \sum_{k \mathop = 0}^{n - 1} 10^k
| c = [[Basis Representation Theorem]]
}}
{{eqn | r = \sum_{j \mathop = 0}^{s - 1} \paren {\sum_{k \mathop = 0}^{r - 1} 10^k} 10^{r j}
| c =
}}
{{eqn | r = \paren {\sum_{k \mathop = 0}^{r - 1} 10^k} \paren {\s... | Divisors of Repunit with Composite Index | https://proofwiki.org/wiki/Divisors_of_Repunit_with_Composite_Index | https://proofwiki.org/wiki/Divisors_of_Repunit_with_Composite_Index | [
"Repunits"
] | [
"Definition:Repunit",
"Definition:Digit",
"Definition:Composite Number",
"Definition:Divisor (Algebra)/Integer"
] | [
"Basis Representation Theorem",
"Category:Repunits"
] |
proofwiki-13802 | Repunit in Base 9 is Triangular | Let $m$ be a repunit base $9$.
Then $m$ is a triangular number. | Let $m$ be a repunit base $9$ with $n$ digits.
We have:
{{begin-eqn}}
{{eqn | l = m
| r = \sum_{k \mathop = 0}^{n - 1} 9^k
| c = Basis Representation Theorem
}}
{{eqn | r = \dfrac {9^n - 1} {9 - 1}
| c = Sum of Geometric Sequence
}}
{{eqn | r = \dfrac {3^{2 n} - 1} 8
| c =
}}
{{eqn | r = \dfrac... | Let $m$ be a [[Definition:Repunit|repunit base $9$]].
Then $m$ is a [[Definition:Triangular Number|triangular number]]. | Let $m$ be a [[Definition:Repunit|repunit base $9$]] with $n$ [[Definition:Digit|digits]].
We have:
{{begin-eqn}}
{{eqn | l = m
| r = \sum_{k \mathop = 0}^{n - 1} 9^k
| c = [[Basis Representation Theorem]]
}}
{{eqn | r = \dfrac {9^n - 1} {9 - 1}
| c = [[Sum of Geometric Sequence]]
}}
{{eqn | r = \df... | Repunit in Base 9 is Triangular | https://proofwiki.org/wiki/Repunit_in_Base_9_is_Triangular | https://proofwiki.org/wiki/Repunit_in_Base_9_is_Triangular | [
"Repunits",
"Triangular Numbers"
] | [
"Definition:Repunit",
"Definition:Triangular Number"
] | [
"Definition:Repunit",
"Definition:Digit",
"Basis Representation Theorem",
"Sum of Geometric Sequence",
"Difference of Two Squares",
"Definition:Odd Integer",
"Definition:Integer",
"Closed Form for Triangular Numbers"
] |
proofwiki-13803 | Square of Small Repunit is Palindromic | The squares of repunits with up to $9$ digits are palindromic. | {{begin-eqn}}
{{eqn | l = 1^2
| r = 1
}}
{{eqn | l = 11^2
| r = 121
| c =
}}
{{eqn | l = 111^2
| r = 12 \, 321
| c =
}}
{{eqn | l = 1111^2
| r = 1 \, 234 \, 321
| c =
}}
{{eqn | l = 11 \, 111^2
| r = 123 \, 454 \, 321
| c =
}}
{{eqn | l = 111 \, 111^2
| r ... | The [[Definition:Square (Algebra)|squares]] of [[Definition:Repunit|repunits]] with up to $9$ [[Definition:Digit|digits]] are [[Definition:Palindromic Number|palindromic]]. | {{begin-eqn}}
{{eqn | l = 1^2
| r = 1
}}
{{eqn | l = 11^2
| r = 121
| c =
}}
{{eqn | l = 111^2
| r = 12 \, 321
| c =
}}
{{eqn | l = 1111^2
| r = 1 \, 234 \, 321
| c =
}}
{{eqn | l = 11 \, 111^2
| r = 123 \, 454 \, 321
| c =
}}
{{eqn | l = 111 \, 111^2
| r ... | Square of Small Repunit is Palindromic | https://proofwiki.org/wiki/Square_of_Small_Repunit_is_Palindromic | https://proofwiki.org/wiki/Square_of_Small_Repunit_is_Palindromic | [
"Repunits",
"Square Numbers"
] | [
"Definition:Square/Function",
"Definition:Repunit",
"Definition:Digit",
"Definition:Palindromic Number"
] | [] |
proofwiki-13804 | 5 Numbers such that Sum of any 3 is Square | This set of $5$ integers has the property that the sum of any $3$ of them is square:
{{begin-eqn}}
{{eqn | l = 26 \, 072 \, 323 \, 311 \, 568 \, 661 \, 931
| o =
}}
{{eqn | l = 43 \, 744 \, 839 \, 742 \, 282 \, 591 \, 947
| o =
}}
{{eqn | l = 118 \, 132 \, 654 \, 413 \, 675 \, 138 \, 222
| o =
}}
{... | Taking the $\dbinom 5 3 = 10$ subsets of $3$ integers at a time:
{{begin-eqn}}
{{eqn | n = 1
| l = 26 \, 072 \, 323 \, 311 \, 568 \, 661 \, 931
| o =
}}
{{eqn | lo= +
| l = 43 \, 744 \, 839 \, 742 \, 282 \, 591 \, 947
| o =
}}
{{eqn | lo= +
| l = 118 \, 132 \, 654 \, 413 \, 675 \, 138 \,... | This [[Definition:Set|set]] of $5$ [[Definition:Integer|integers]] has the property that the [[Definition:Integer Addition|sum]] of any $3$ of them is [[Definition:Square Number|square]]:
{{begin-eqn}}
{{eqn | l = 26 \, 072 \, 323 \, 311 \, 568 \, 661 \, 931
| o =
}}
{{eqn | l = 43 \, 744 \, 839 \, 742 \, 282 \... | Taking the $\dbinom 5 3 = 10$ [[Definition:Subset|subsets]] of $3$ [[Definition:Integer|integers]] at a time:
{{begin-eqn}}
{{eqn | n = 1
| l = 26 \, 072 \, 323 \, 311 \, 568 \, 661 \, 931
| o =
}}
{{eqn | lo= +
| l = 43 \, 744 \, 839 \, 742 \, 282 \, 591 \, 947
| o =
}}
{{eqn | lo= +
... | 5 Numbers such that Sum of any 3 is Square | https://proofwiki.org/wiki/5_Numbers_such_that_Sum_of_any_3_is_Square | https://proofwiki.org/wiki/5_Numbers_such_that_Sum_of_any_3_is_Square | [
"Square Numbers"
] | [
"Definition:Set",
"Definition:Integer",
"Definition:Addition/Integers",
"Definition:Square Number"
] | [
"Definition:Subset",
"Definition:Integer"
] |
proofwiki-13805 | Largest nth Power which has n Digits | The largest $n$th power which has $n$ digits is $9^{21}$:
:$9^{21} = 109 \, 418 \, 989 \, 131 \, 512 \, 359 \, 209$ | The $n$th power of $10$ has $n + 1$ digits.
Hence the $n$th power of $m$ such that $m > 10$ has more than $n$ digits.
The $11$th power of $8$ has $10$ digits:
:$8^{10} = 8 \, 589 \, 934 \, 592$
and so $8^n$ where $n > 10$ has fewer than $n$ digits.
Hence the $n$th power of $m$ such that $m < 9$ and $n < 21$ has fewer t... | The largest [[Definition:Integer Power|$n$th power]] which has $n$ [[Definition:Digit|digits]] is $9^{21}$:
:$9^{21} = 109 \, 418 \, 989 \, 131 \, 512 \, 359 \, 209$ | The $n$th power of $10$ has $n + 1$ [[Definition:Digit|digits]].
Hence the $n$th [[Definition:Integer Power|power]] of $m$ such that $m > 10$ has more than $n$ [[Definition:Digit|digits]].
The $11$th [[Definition:Integer Power|power]] of $8$ has $10$ [[Definition:Digit|digits]]:
:$8^{10} = 8 \, 589 \, 934 \, 592$
an... | Largest nth Power which has n Digits | https://proofwiki.org/wiki/Largest_nth_Power_which_has_n_Digits | https://proofwiki.org/wiki/Largest_nth_Power_which_has_n_Digits | [
"Powers",
"Recreational Mathematics",
"109,418,989,131,512,359,209"
] | [
"Definition:Power (Algebra)/Integer",
"Definition:Digit"
] | [
"Definition:Digit",
"Definition:Power (Algebra)/Integer",
"Definition:Digit",
"Definition:Power (Algebra)/Integer",
"Definition:Digit",
"Definition:Digit",
"Definition:Power (Algebra)/Integer",
"Definition:Digit",
"Definition:Power (Algebra)/Integer",
"Definition:Digit"
] |
proofwiki-13806 | Smallest Number which is Multiplied by 99 by Appending 1 to Each End | The smallest positive integer which is multiplied by $99$ when $1$ is appended to each end is:
:$112 \, 359 \, 550 \, 561 \, 797 \, 752 \, 809$ | We have that:
:$112 \, 359 \, 550 \, 561 \, 797 \, 752 \, 809 = 101 \times 1 \, 052 \, 788 \, 969 \times 1 \, 056 \, 689 \, 261$
while:
{{begin-eqn}}
{{eqn | l = 11 \, 123 \, 595 \, 505 \, 617 \, 977 \, 528 \, 091
| r = 3^2 \times 11 \times 101 \times 1 \, 052 \, 788 \, 969 \times 1 \, 056 \, 689 \, 261
| c... | The smallest [[Definition:Positive Integer|positive integer]] which is multiplied by $99$ when $1$ is appended to each end is:
:$112 \, 359 \, 550 \, 561 \, 797 \, 752 \, 809$ | We have that:
:$112 \, 359 \, 550 \, 561 \, 797 \, 752 \, 809 = 101 \times 1 \, 052 \, 788 \, 969 \times 1 \, 056 \, 689 \, 261$
while:
{{begin-eqn}}
{{eqn | l = 11 \, 123 \, 595 \, 505 \, 617 \, 977 \, 528 \, 091
| r = 3^2 \times 11 \times 101 \times 1 \, 052 \, 788 \, 969 \times 1 \, 056 \, 689 \, 261
... | Smallest Number which is Multiplied by 99 by Appending 1 to Each End | https://proofwiki.org/wiki/Smallest_Number_which_is_Multiplied_by_99_by_Appending_1_to_Each_End | https://proofwiki.org/wiki/Smallest_Number_which_is_Multiplied_by_99_by_Appending_1_to_Each_End | [
"Recreational Mathematics",
"112,359,550,561,797,752,809"
] | [
"Definition:Positive/Integer"
] | [
"Definition:Integer",
"Definition:Digit",
"Definition:Divisor (Algebra)/Integer"
] |
proofwiki-13807 | Smallest Multiply Perfect Number of Order 6 | The number $154 \, 345 \, 556 \, 085 \, 770 \, 649 \, 600$ is multiply perfect of order $6$:
{{begin-eqn}}
{{eqn | l = \map {\sigma_1} {154 \, 345 \, 556 \, 085 \, 770 \, 649 \, 600}
| r = 926 \, 073 \, 336 \, 514 \, 623 \, 897 \, 600
| c =
}}
{{eqn | r = 6 \times 154 \, 345 \, 556 \, 085 \, 770 \, 649 \, ... | From {{DSFLink|154,345,556,085,770,649,600|154 \, 345 \, 556 \, 085 \, 770 \, 649 \, 600}}:
:$\map {\sigma_1} {154 \, 345 \, 556 \, 085 \, 770 \, 649 \, 600} = 926 \, 073 \, 336 \, 514 \, 623 \, 897 \, 600$
{{ProofWanted|That it is the smallest one remains to be proved.}} | The number $154 \, 345 \, 556 \, 085 \, 770 \, 649 \, 600$ is [[Definition:Multiply Perfect Number|multiply perfect]] of [[Definition:Order of Multiply Perfect Number|order]] $6$:
{{begin-eqn}}
{{eqn | l = \map {\sigma_1} {154 \, 345 \, 556 \, 085 \, 770 \, 649 \, 600}
| r = 926 \, 073 \, 336 \, 514 \, 623 \, 89... | From {{DSFLink|154,345,556,085,770,649,600|154 \, 345 \, 556 \, 085 \, 770 \, 649 \, 600}}:
:$\map {\sigma_1} {154 \, 345 \, 556 \, 085 \, 770 \, 649 \, 600} = 926 \, 073 \, 336 \, 514 \, 623 \, 897 \, 600$
{{ProofWanted|That it is the smallest one remains to be proved.}} | Smallest Multiply Perfect Number of Order 6 | https://proofwiki.org/wiki/Smallest_Multiply_Perfect_Number_of_Order_6 | https://proofwiki.org/wiki/Smallest_Multiply_Perfect_Number_of_Order_6 | [
"Multiply Perfect Numbers",
"154,345,556,085,770,649,600"
] | [
"Definition:Multiply Perfect Number",
"Definition:Multiply Perfect Number/Order",
"Definition:Positive/Integer"
] | [] |
proofwiki-13808 | Sequence of 9 Consecutive Integers each with 48 Divisors | The $9$ integers beginning $17 \, 796 \, 126 \, 877 \, 482 \, 329 \, 126 \, 044$ each has $48$ divisors. | In the below, $\sigma_0$ denotes the divisor count function.
{{begin-eqn}}
{{eqn | l = \map {\sigma_0} {17 \, 796 \, 126 \, 877 \, 482 \, 329 \, 126 \, 044}
| r = 48
| c = {{DCFLink|17,796,126,877,482,329,126,044|17 \, 796 \, 126 \, 877 \, 482 \, 329 \, 126 \, 044}}
}}
{{eqn | l = \map {\sigma_0} {17 \, 796... | The $9$ [[Definition:Integer|integers]] beginning $17 \, 796 \, 126 \, 877 \, 482 \, 329 \, 126 \, 044$ each has $48$ [[Definition:Divisor of Integer|divisors]]. | In the below, $\sigma_0$ denotes the [[Definition:Divisor Count Function|divisor count function]].
{{begin-eqn}}
{{eqn | l = \map {\sigma_0} {17 \, 796 \, 126 \, 877 \, 482 \, 329 \, 126 \, 044}
| r = 48
| c = {{DCFLink|17,796,126,877,482,329,126,044|17 \, 796 \, 126 \, 877 \, 482 \, 329 \, 126 \, 044}}
}... | Sequence of 9 Consecutive Integers each with 48 Divisors | https://proofwiki.org/wiki/Sequence_of_9_Consecutive_Integers_each_with_48_Divisors | https://proofwiki.org/wiki/Sequence_of_9_Consecutive_Integers_each_with_48_Divisors | [
"Divisor Count Function"
] | [
"Definition:Integer",
"Definition:Divisor (Algebra)/Integer"
] | [
"Definition:Divisor Count Function"
] |
proofwiki-13809 | Left-Truncatable Prime/Examples/357,686,312,646,216,567,629,137 | The largest left-truncatable prime is $357 \, 686 \, 312 \, 646 \, 216 \, 567 \, 629 \, 137$. | First it is demonstrated that $357 \, 686 \, 312 \, 646 \, 216 \, 567 \, 629 \, 137$ is indeed a left-truncatable prime:
{{begin-eqn}}
{{eqn | o =
| l = 357 \, 686 \, 312 \, 646 \, 216 \, 567 \, 629 \, 137
| c = is prime
}}
{{eqn | o =
| l = 57 \, 686 \, 312 \, 646 \, 216 \, 567 \, 629 \, 137
... | The largest [[Definition:Left-Truncatable Prime|left-truncatable prime]] is $357 \, 686 \, 312 \, 646 \, 216 \, 567 \, 629 \, 137$. | First it is demonstrated that $357 \, 686 \, 312 \, 646 \, 216 \, 567 \, 629 \, 137$ is indeed a [[Definition:Left-Truncatable Prime|left-truncatable prime]]:
{{begin-eqn}}
{{eqn | o =
| l = 357 \, 686 \, 312 \, 646 \, 216 \, 567 \, 629 \, 137
| c = is [[Definition:Prime Number|prime]]
}}
{{eqn | o =
... | Left-Truncatable Prime/Examples/357,686,312,646,216,567,629,137 | https://proofwiki.org/wiki/Left-Truncatable_Prime/Examples/357,686,312,646,216,567,629,137 | https://proofwiki.org/wiki/Left-Truncatable_Prime/Examples/357,686,312,646,216,567,629,137 | [
"Left-Truncatable Primes"
] | [
"Definition:Left-Truncatable Prime"
] | [
"Definition:Left-Truncatable Prime",
"Definition:Prime Number",
"Definition:Prime Number",
"Definition:Prime Number",
"Definition:Prime Number",
"Definition:Prime Number",
"Definition:Prime Number",
"Definition:Prime Number",
"Definition:Prime Number",
"Definition:Prime Number",
"Definition:Prim... |
proofwiki-13810 | Polydivisible Number/Examples/3,608,528,850,368,400,786,036,725 | The largest polydivisible number has $25$ digits:
:$3 \, 608 \, 528 \, 850 \, 368 \, 400 \, 786 \, 036 \, 725$ | {{begin-eqn}}
{{eqn | l = 3
| r = 1 \times 3
}}
{{eqn | l = 36
| r = 2 \times 18
}}
{{eqn | l = 360
| r = 3 \times 120
}}
{{eqn | l = 3608
| r = 4 \times 902
}}
{{eqn | l = 36 \, 085
| r = 5 \times 7217
}}
{{eqn | l = 360 \, 852
| r = 6 \times 60 \, 142
}}
{{eqn | l = 3 \, 608 \, 528... | The largest [[Definition:Polydivisible Number|polydivisible number]] has $25$ [[Definition:Digit|digits]]:
:$3 \, 608 \, 528 \, 850 \, 368 \, 400 \, 786 \, 036 \, 725$ | {{begin-eqn}}
{{eqn | l = 3
| r = 1 \times 3
}}
{{eqn | l = 36
| r = 2 \times 18
}}
{{eqn | l = 360
| r = 3 \times 120
}}
{{eqn | l = 3608
| r = 4 \times 902
}}
{{eqn | l = 36 \, 085
| r = 5 \times 7217
}}
{{eqn | l = 360 \, 852
| r = 6 \times 60 \, 142
}}
{{eqn | l = 3 \, 608 \, 528... | Polydivisible Number/Examples/3,608,528,850,368,400,786,036,725 | https://proofwiki.org/wiki/Polydivisible_Number/Examples/3,608,528,850,368,400,786,036,725 | https://proofwiki.org/wiki/Polydivisible_Number/Examples/3,608,528,850,368,400,786,036,725 | [
"Polydivisible Numbers",
"3,608,528,850,368,400,786,036,725"
] | [
"Definition:Polydivisible Number",
"Definition:Digit"
] | [
"No Polydivisible Number with 26 Digits Exists",
"Definition:Polydivisible Number"
] |
proofwiki-13811 | Probability of All Players receiving Complete Suit at Bridge | The probability of all $4$ players in a game of Bridge being dealt a complete suit is $1$ in $2 \, 235 \, 197 \, 406 \, 895 \, 366 \, 368 \, 301 \, 560 \, 000$. | {{ProofWanted|Straightforward but boring exercise in combinatorics}} | The [[Definition:Probability|probability]] of all $4$ players in a game of [[Definition:Bridge (Game)|Bridge]] being dealt a complete [[Definition:Suit of Cards|suit]] is $1$ in $2 \, 235 \, 197 \, 406 \, 895 \, 366 \, 368 \, 301 \, 560 \, 000$. | {{ProofWanted|Straightforward but boring exercise in combinatorics}} | Probability of All Players receiving Complete Suit at Bridge | https://proofwiki.org/wiki/Probability_of_All_Players_receiving_Complete_Suit_at_Bridge | https://proofwiki.org/wiki/Probability_of_All_Players_receiving_Complete_Suit_at_Bridge | [
"Bridge (Game)"
] | [
"Definition:Probability",
"Definition:Bridge (Game)",
"Definition:Deck of Cards/Suit"
] | [] |
proofwiki-13812 | Powers of 10 Expressible as Product of 2 Zero-Free Factors | The powers of $10$ which can be expressed as the product of $2$ factors neither of which has a zero in its decimal representation are:
{{begin-eqn}}
{{eqn | l = 10^1
| r = 2 \times 5
| c =
}}
{{eqn | l = 10^2
| r = 4 \times 25
}}
{{eqn | l = 10^3
| r = 8 \times 125
}}
{{eqn | l = 10^4
| r... | Let $p q = 10^n$ for some $n \in \Z_{>0}$.
Then $p q = 2^n 5^n$.
{{WLOG}}, suppose $p = 2^r 5^s$ for $r, s \ge 1$.
Then $2 \times 5 = 10$ is a divisor of $p$ and so $p$ ends with a zero.
Thus for $p$ and $q$ to be zero-free, it must be the case that $p = 2^n$ and $q = 5^n$ (or the other way around).
The result follows ... | The [[Definition:Integer Power|powers]] of $10$ which can be expressed as the [[Definition:Integer Multiplication|product]] of $2$ [[Definition:Divisor of Integer|factors]] neither of which has a [[Definition:Zero Digit|zero]] in its [[Definition:Decimal Notation|decimal representation]] are:
{{begin-eqn}}
{{eqn | l =... | Let $p q = 10^n$ for some $n \in \Z_{>0}$.
Then $p q = 2^n 5^n$.
{{WLOG}}, suppose $p = 2^r 5^s$ for $r, s \ge 1$.
Then $2 \times 5 = 10$ is a [[Definition:Divisor of Integer|divisor]] of $p$ and so $p$ ends with a [[Definition:Zero Digit|zero]].
Thus for $p$ and $q$ to be [[Definition:Zero Digit|zero]]-free, it mu... | Powers of 10 Expressible as Product of 2 Zero-Free Factors | https://proofwiki.org/wiki/Powers_of_10_Expressible_as_Product_of_2_Zero-Free_Factors | https://proofwiki.org/wiki/Powers_of_10_Expressible_as_Product_of_2_Zero-Free_Factors | [
"Powers of 10"
] | [
"Definition:Power (Algebra)/Integer",
"Definition:Multiplication/Integers",
"Definition:Divisor (Algebra)/Integer",
"Definition:Zero Digit",
"Definition:Decimal Notation"
] | [
"Definition:Divisor (Algebra)/Integer",
"Definition:Zero Digit",
"Definition:Zero Digit",
"Powers of 2 and 5 without Zeroes"
] |
proofwiki-13813 | Sequence of Palindromic Sophie Germain Primes | The number $N = 191 \, 918 \, 080 \, 818 \, 091 \, 909 \, 090 \, 909 \, 190 \, 818 \, 080 \, 819 \, 191$ has the property that:
:$N$ is a palindromic Sophie Germain prime
:$2 N + 1$ is also a palindromic Sophie Germain prime
:$2 \left({2 N + 1}\right) + 1$ is also a palindromic prime, but not a Sophie Germain prime. | By direct calculation:
{{begin-eqn}}
{{eqn | o =
| r = 191 \, 918 \, 080 \, 818 \, 091 \, 909 \, 090 \, 909 \, 190 \, 818 \, 080 \, 819 \, 191
| c = is palindromic and prime
}}
{{eqn | o =
| r = 2 \times 191 \, 918 \, 080 \, 818 \, 091 \, 909 \, 090 \, 909 \, 190 \, 818 \, 080 \, 819 \, 191 + 1
... | The number $N = 191 \, 918 \, 080 \, 818 \, 091 \, 909 \, 090 \, 909 \, 190 \, 818 \, 080 \, 819 \, 191$ has the property that:
:$N$ is a [[Definition:Palindromic Prime|palindromic]] [[Definition:Sophie Germain Prime|Sophie Germain prime]]
:$2 N + 1$ is also a [[Definition:Palindromic Prime|palindromic]] [[Definition... | By direct calculation:
{{begin-eqn}}
{{eqn | o =
| r = 191 \, 918 \, 080 \, 818 \, 091 \, 909 \, 090 \, 909 \, 190 \, 818 \, 080 \, 819 \, 191
| c = is [[Definition:Palindromic Prime|palindromic]] and [[Definition:Prime Number|prime]]
}}
{{eqn | o =
| r = 2 \times 191 \, 918 \, 080 \, 818 \, 091 \, ... | Sequence of Palindromic Sophie Germain Primes | https://proofwiki.org/wiki/Sequence_of_Palindromic_Sophie_Germain_Primes | https://proofwiki.org/wiki/Sequence_of_Palindromic_Sophie_Germain_Primes | [
"Sophie Germain Primes",
"Palindromic Primes"
] | [
"Definition:Palindromic Prime",
"Definition:Sophie Germain Prime",
"Definition:Palindromic Prime",
"Definition:Sophie Germain Prime",
"Definition:Palindromic Prime",
"Definition:Prime Number",
"Definition:Sophie Germain Prime"
] | [
"Definition:Palindromic Prime",
"Definition:Prime Number",
"Definition:Palindromic Prime",
"Definition:Prime Number",
"Definition:Palindromic Prime",
"Definition:Prime Number",
"Definition:Prime Number",
"Definition:Palindromic Prime"
] |
proofwiki-13814 | Smallest Integer Divisible by All Numbers from 1 to 100 | The smallest positive integer which is divisible by each of the integers from $1$ to $100$ is:
:$69 \, 720 \, 375 \, 229 \, 712 \, 477 \, 164 \, 533 \, 808 \, 935 \, 312 \, 303 \, 556 \, 800$ | Let $N$ be divisible by each of the integers from $1$ to $100$.
Each prime number between $2$ and $97$ must be a divisor of $N$.
Also:
:$2^6 = 64 \divides N$
:$3^4 = 81 \divides N$
:$5^2 = 25 \divides N$
:$7^2 = 49 \divides N$
Every other integer between $1$ and $100$ is the product of a subset of all of these.
Hence b... | The smallest [[Definition:Positive Integer|positive integer]] which is [[Definition:Divisor of Integer|divisible]] by each of the [[Definition:Integer|integers]] from $1$ to $100$ is:
:$69 \, 720 \, 375 \, 229 \, 712 \, 477 \, 164 \, 533 \, 808 \, 935 \, 312 \, 303 \, 556 \, 800$ | Let $N$ be [[Definition:Divisor of Integer|divisible]] by each of the [[Definition:Integer|integers]] from $1$ to $100$.
Each [[Definition:Prime Number|prime number]] between $2$ and $97$ must be a [[Definition:Divisor of Integer|divisor]] of $N$.
Also:
:$2^6 = 64 \divides N$
:$3^4 = 81 \divides N$
:$5^2 = 25 \divi... | Smallest Integer Divisible by All Numbers from 1 to 100 | https://proofwiki.org/wiki/Smallest_Integer_Divisible_by_All_Numbers_from_1_to_100 | https://proofwiki.org/wiki/Smallest_Integer_Divisible_by_All_Numbers_from_1_to_100 | [
"Divisors",
"69,720,375,229,712,477,164,533,808,935,312,303,556,800"
] | [
"Definition:Positive/Integer",
"Definition:Divisor (Algebra)/Integer",
"Definition:Integer"
] | [
"Definition:Divisor (Algebra)/Integer",
"Definition:Integer",
"Definition:Prime Number",
"Definition:Divisor (Algebra)/Integer",
"Definition:Integer",
"Definition:Multiplication/Integers",
"Definition:Subset",
"Euclid's Lemma"
] |
proofwiki-13815 | Integer which is Multiplied by 9 when moving Last Digit to First | Let $N$ be the positive integer:
:$N = 10 \, 112 \, 359 \, 550 \, 561 \, 797 \, 752 \, 808 \, 988 \, 764 \, 044 \, 943 \, 820 \, 224 \, 719$
$N$ is the smallest positive integer $N$ such that if you move the last digit to the front, the result is the positive integer $9 N$. | From Integer which is Multiplied by Last Digit when moving Last Digit to First, $N$ is equal to the recurring part of the fraction:
:$q = \dfrac {a_1} {10 a_1 - 1}$
where $a_1 = 9$.
Thus:
:$q = \dfrac 9 {10 \times 9 - 1} = \dfrac 9 {89}$
Hence:
=== Decimal Expansion ===
{{:Integer which is Multiplied by 9 when moving L... | Let $N$ be the [[Definition:Positive Integer|positive integer]]:
:$N = 10 \, 112 \, 359 \, 550 \, 561 \, 797 \, 752 \, 808 \, 988 \, 764 \, 044 \, 943 \, 820 \, 224 \, 719$
$N$ is the smallest [[Definition:Positive Integer|positive integer]] $N$ such that if you move the last [[Definition:Digit|digit]] to the front, t... | From [[Integer which is Multiplied by Last Digit when moving Last Digit to First]], $N$ is equal to the [[Definition:Recurring Part|recurring part]] of the [[Definition:Fraction|fraction]]:
:$q = \dfrac {a_1} {10 a_1 - 1}$
where $a_1 = 9$.
Thus:
:$q = \dfrac 9 {10 \times 9 - 1} = \dfrac 9 {89}$
Hence:
=== [[Integ... | Integer which is Multiplied by 9 when moving Last Digit to First/Proof 2 | https://proofwiki.org/wiki/Integer_which_is_Multiplied_by_9_when_moving_Last_Digit_to_First | https://proofwiki.org/wiki/Integer_which_is_Multiplied_by_9_when_moving_Last_Digit_to_First/Proof_2 | [
"Integer which is Multiplied by 9 when moving Last Digit to First",
"Recreational Mathematics"
] | [
"Definition:Positive/Integer",
"Definition:Positive/Integer",
"Definition:Digit",
"Definition:Positive/Integer"
] | [
"Integer which is Multiplied by Last Digit when moving Last Digit to First",
"Definition:Basis Expansion/Recurrence/Recurring Part",
"Definition:Fraction",
"Integer which is Multiplied by 9 when moving Last Digit to First/Decimal Expansion"
] |
proofwiki-13816 | Power of 2 containing no Digit 2 | $2^{168}$ contains no $2$ anywhere in its decimal representation. | :$2^{168} = 374 \, 144 \, 419 \, 156 \, 711 \, 147 \, 060 \, 143 \, 317 \, 175 \, 368 \, 453 \, 031 \, 918 \, 731 \, 001 \, 856$
{{qed}} | $2^{168}$ contains no $2$ anywhere in its [[Definition:Decimal Notation|decimal representation]]. | :$2^{168} = 374 \, 144 \, 419 \, 156 \, 711 \, 147 \, 060 \, 143 \, 317 \, 175 \, 368 \, 453 \, 031 \, 918 \, 731 \, 001 \, 856$
{{qed}} | Power of 2 containing no Digit 2 | https://proofwiki.org/wiki/Power_of_2_containing_no_Digit_2 | https://proofwiki.org/wiki/Power_of_2_containing_no_Digit_2 | [
"374,144,419,156,711,147,060,143,317,175,368,453,031,918,731,001,856",
"Powers of 2"
] | [
"Definition:Decimal Notation"
] | [] |
proofwiki-13817 | Smallest Cube whose Sum of Divisors is Cube | The smallest cube $N$ such that $\map {\sigma_1} N$ is also a cube is:
:$27 \, 418 \, 521 \, 963 \, 671 \, 501 \, 273 \, 905 \, 190 \, 135 \, 082 \, 692 \, 041 \, 730 \, 405 \, 303 \, 870 \, 249 \, 023 \, 209$
where $\map {\sigma_1} N$ denotes the divisor sum of $N$. | We have that:
{{begin-eqn}}
{{eqn | l = N
| r = 27 \, 418 \, 521 \, 963 \, 671 \, 501 \, 273 \, 905 \, 190 \, 135 \, 082 \, 692 \, 041 \, 730 \, 405 \, 303 \, 870 \, 249 \, 023 \, 209
| c =
}}
{{eqn | r = 3^9 \times 7^3 \times 11^3 \times 13^3 \times 17^3 \times 41^3 \times 43^3 \times 47^3 \times 443^3 \t... | The smallest [[Definition:Cube Number|cube]] $N$ such that $\map {\sigma_1} N$ is also a [[Definition:Cube Number|cube]] is:
:$27 \, 418 \, 521 \, 963 \, 671 \, 501 \, 273 \, 905 \, 190 \, 135 \, 082 \, 692 \, 041 \, 730 \, 405 \, 303 \, 870 \, 249 \, 023 \, 209$
where $\map {\sigma_1} N$ denotes the [[Definition:Divis... | We have that:
{{begin-eqn}}
{{eqn | l = N
| r = 27 \, 418 \, 521 \, 963 \, 671 \, 501 \, 273 \, 905 \, 190 \, 135 \, 082 \, 692 \, 041 \, 730 \, 405 \, 303 \, 870 \, 249 \, 023 \, 209
| c =
}}
{{eqn | r = 3^9 \times 7^3 \times 11^3 \times 13^3 \times 17^3 \times 41^3 \times 43^3 \times 47^3 \times 443^3 \... | Smallest Cube whose Sum of Divisors is Cube | https://proofwiki.org/wiki/Smallest_Cube_whose_Sum_of_Divisors_is_Cube | https://proofwiki.org/wiki/Smallest_Cube_whose_Sum_of_Divisors_is_Cube | [
"Cube Numbers",
"Divisor Sum Function"
] | [
"Definition:Cube Number",
"Definition:Cube Number",
"Definition:Divisor Sum Function"
] | [] |
proofwiki-13818 | Uniqueness of Polynomial Ring in One Variable | Let $R$ be a commutative ring with unity.
Let $\struct {R \sqbrk X, \iota, X}$ and $\struct {R \sqbrk Y, \kappa, Y}$ be polynomial rings in one variable over $R$.
Then there exists a unique ring homomorphism $f: R \sqbrk X \to R \sqbrk Y$ such that:
:$f \circ \iota = \kappa$
:$\map f X = Y$
and it is an isomorphism. | The existence and uniqueness of $f$ follows from the universal property.
Likewise, there is a unique ring homomorphism $g: R \sqbrk Y \to R \sqbrk X$ such that:
:$g \circ \kappa = \iota$
:$\map g Y = X$
and a unique ring homomorphism $h: R \sqbrk X \to R \sqbrk X$ such that:
:$h \circ \iota = \iota$
:$\map X = X$
By un... | Let $R$ be a [[Definition:Commutative Ring with Unity|commutative ring with unity]].
Let $\struct {R \sqbrk X, \iota, X}$ and $\struct {R \sqbrk Y, \kappa, Y}$ be [[Definition:Polynomial Ring in One Variable|polynomial rings in one variable]] over $R$.
Then there exists a [[Definition:Unique|unique]] [[Definition:Ri... | The existence and uniqueness of $f$ follows from the [[Definition:Universal Property of Polynomial Ring in One Variable|universal property]].
Likewise, there is a unique [[Definition:Ring Homomorphism|ring homomorphism]] $g: R \sqbrk Y \to R \sqbrk X$ such that:
:$g \circ \kappa = \iota$
:$\map g Y = X$
and a unique [... | Uniqueness of Polynomial Ring in One Variable | https://proofwiki.org/wiki/Uniqueness_of_Polynomial_Ring_in_One_Variable | https://proofwiki.org/wiki/Uniqueness_of_Polynomial_Ring_in_One_Variable | [
"Polynomial Theory"
] | [
"Definition:Commutative and Unitary Ring",
"Definition:Polynomial Ring",
"Definition:Unique",
"Definition:Ring Homomorphism",
"Definition:Isomorphism (Abstract Algebra)/Ring Isomorphism"
] | [
"Definition:Polynomial Ring/Universal Property",
"Definition:Ring Homomorphism",
"Definition:Ring Homomorphism",
"Identity Mapping is Ring Homomorphism",
"Definition:Identity Mapping",
"Composition of Ring Homomorphisms is Ring Homomorphism",
"Definition:Isomorphism (Abstract Algebra)/Ring Isomorphism",... |
proofwiki-13819 | Monomials of Polynomial Ring are Linearly Independent/One Variable | Let $R$ be a commutative ring with unity.
Let $R \sqbrk X$ be a polynomial ring in one variable $X$ over $R$.
Then the set of monomials $\set {X^k : k \in \N}$ is linearly independent. | We consider polynomial ring over sequences in $R$:
Let $\struct {R^{\left({\N}\right)}, \oplus, \odot}$ be the underlying ring of sequences of finite support.
Recall:
:$X = \sequence {0, 1, 0, 0, \ldots}$
Observe:
{{begin-eqn}}
{{eqn | l = X^2
| r = \sequence {0, 0, 1, 0, 0, \ldots}
}}
{{eqn | l = X^3
| r =... | Let $R$ be a [[Definition:Commutative Ring with Unity|commutative ring with unity]].
Let $R \sqbrk X$ be a [[Definition:Polynomial Ring in One Variable|polynomial ring in one variable]] $X$ over $R$.
Then the [[Definition:Set|set]] of [[Definition:Monomial of Polynomial Ring|monomials]] $\set {X^k : k \in \N}$ is [[D... | We consider [[Definition:Polynomial Ring over Sequences|polynomial ring over sequences in $R$]]:
Let $\struct {R^{\left({\N}\right)}, \oplus, \odot}$ be the underlying [[Definition:Ring of Sequences of Finite Support|ring of sequences of finite support]].
Recall:
:$X = \sequence {0, 1, 0, 0, \ldots}$
Observe:
{{be... | Monomials of Polynomial Ring are Linearly Independent/One Variable | https://proofwiki.org/wiki/Monomials_of_Polynomial_Ring_are_Linearly_Independent/One_Variable | https://proofwiki.org/wiki/Monomials_of_Polynomial_Ring_are_Linearly_Independent/One_Variable | [
"Monomials"
] | [
"Definition:Commutative and Unitary Ring",
"Definition:Polynomial Ring",
"Definition:Set",
"Definition:Monomial of Polynomial Ring",
"Definition:Linearly Independent/Set"
] | [
"Definition:Polynomial Ring/Sequences",
"Definition:Ring of Sequences of Finite Support"
] |
proofwiki-13820 | Equivalence of Definitions of Constant Polynomial | Let $R$ be a commutative ring with unity.
Let $P\in R[x]$ be a polynomial in one variable over $R$.
{{TFAE|def = Constant Polynomial}} | === 1 iff 2 ===
This is by definition of coefficients and degree.
{{qed|lemma}} | Let $R$ be a [[Definition:Commutative Ring with Unity|commutative ring with unity]].
Let $P\in R[x]$ be a [[Definition:Polynomial over Ring in One Variable|polynomial in one variable]] over $R$.
{{TFAE|def = Constant Polynomial}} | === 1 iff 2 ===
This is by definition of [[Definition:Coefficient of Polynomial|coefficients]] and [[Definition:Degree of Polynomial|degree]].
{{qed|lemma}} | Equivalence of Definitions of Constant Polynomial | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Constant_Polynomial | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Constant_Polynomial | [
"Constant Polynomials"
] | [
"Definition:Commutative and Unitary Ring",
"Definition:Polynomial over Ring/One Variable"
] | [
"Definition:Coefficient of Polynomial",
"Definition:Degree of Polynomial"
] |
proofwiki-13821 | Equivalence of Definitions of Minimal Polynomial | {{TFAE|def = Minimal Polynomial}}
Let $L / K$ be a field extension.
Let $\alpha \in L$ be algebraic over $K$. | === 1 equals 2 ===
By Minimal Polynomial is Irreducible, it follows that the two are equal.
{{qed|lemma}} | {{TFAE|def = Minimal Polynomial}}
Let $L / K$ be a [[Definition:Field Extension|field extension]].
Let $\alpha \in L$ be [[Definition:Algebraic Element of Field Extension|algebraic]] over $K$. | === 1 equals 2 ===
By [[Minimal Polynomial is Irreducible]], it follows that the two are equal.
{{qed|lemma}} | Equivalence of Definitions of Minimal Polynomial | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Minimal_Polynomial | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Minimal_Polynomial | [
"Minimal Polynomials"
] | [
"Definition:Field Extension",
"Definition:Algebraic Element of Field Extension"
] | [
"Minimal Polynomial is Irreducible"
] |
proofwiki-13822 | Nonzero Ideal of Polynomial Ring over Field has Unique Monic Generator | Let $K$ be a field.
Let $K \sqbrk x$ be the polynomial ring in one variable over $K$.
Let $I \subseteq K \sqbrk x$ be a nonzero ideal.
Then $I$ is generated by a unique monic polynomial. | {{proof wanted}}
Category:Polynomial Rings
damy55ww2yb2hn4pb8w6udwm6porpux | Let $K$ be a [[Definition:Field (Abstract Algebra)|field]].
Let $K \sqbrk x$ be the [[Definition:Polynomial Ring in One Variable|polynomial ring in one variable]] over $K$.
Let $I \subseteq K \sqbrk x$ be a [[Definition:Nonzero Ideal of Ring|nonzero ideal]].
Then $I$ is [[Definition:Generator of Ideal|generated]] b... | {{proof wanted}}
[[Category:Polynomial Rings]]
damy55ww2yb2hn4pb8w6udwm6porpux | Nonzero Ideal of Polynomial Ring over Field has Unique Monic Generator | https://proofwiki.org/wiki/Nonzero_Ideal_of_Polynomial_Ring_over_Field_has_Unique_Monic_Generator | https://proofwiki.org/wiki/Nonzero_Ideal_of_Polynomial_Ring_over_Field_has_Unique_Monic_Generator | [
"Polynomial Rings"
] | [
"Definition:Field (Abstract Algebra)",
"Definition:Polynomial Ring",
"Definition:Non-Null Ideal",
"Definition:Generator of Ideal of Ring",
"Definition:Unique",
"Definition:Monic Polynomial"
] | [
"Category:Polynomial Rings"
] |
proofwiki-13823 | Equivalence of Definitions of Integral Element of Algebra | {{TFAE|def = Integral Element of Algebra}}
Let $A$ be a commutative ring with unity.
Let $f : A \to B$ be a commutative $A$-algebra.
Let $b\in B$. | === Definition 1 implies Definition 2 ===
Assume $b$ is a root of a monic polynomial in $A \sqbrk x$.
That is, there are $n \in \N_{>0}$ and $a_1, \ldots , a_{n-1} \in A$ be such that:
:$b^n + a_{n - 1} b^{n-1} + \cdots + a_1 b + a_0 = 0$
That is:
:$b^n = - a_0 - a_1 b - \cdots - a_{n-1} b^{n-1}$
Therefore $\set {1, b,... | {{TFAE|def = Integral Element of Algebra}}
Let $A$ be a [[Definition:Commutative Ring with Unity|commutative ring with unity]].
Let $f : A \to B$ be a [[Definition:Unital Associative Commutative Algebra|commutative $A$-algebra]].
Let $b\in B$. | === [[Definition:Integral Element of Algebra/Definition 1|Definition 1]] implies [[Definition:Integral Element of Algebra/Definition 2|Definition 2]] ===
Assume $b$ is a [[Definition:Root of Polynomial|root]] of a [[Definition:Monic Polynomial|monic polynomial]] in $A \sqbrk x$.
That is, there are $n \in \N_{>0}$ and... | Equivalence of Definitions of Integral Element of Algebra | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Integral_Element_of_Algebra | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Integral_Element_of_Algebra | [
"Integral Elements"
] | [
"Definition:Commutative and Unitary Ring",
"Definition:Unital Associative Commutative Algebra"
] | [
"Definition:Integral Element of Algebra/Definition 1",
"Definition:Integral Element of Algebra/Definition 2",
"Definition:Root of Polynomial",
"Definition:Monic Polynomial",
"Definition:Generator of Module",
"Definition:Integral Element of Algebra/Definition 1",
"Definition:Integral Element of Algebra/D... |
proofwiki-13824 | Product of Closed Sets is Closed | Let $\family {\struct {S_i, \tau_i} }_{i \mathop \in I}$ be a family of topological spaces where $I$ is an arbitrary indexing set.
Let $\ds S = \prod_{i \mathop \in I} S_i$.
Let $T = \struct {S, \TT}$ be the product space of $\family {\struct {S_i, \tau_i} }_{i \mathop \in I}$ with the product topology $\TT$.
Suppose w... | First note that:
{{begin-eqn}}
{{eqn | l = \prod_{i \mathop \in I} C_i
| r = \set {x \in S : \forall i \in I, x_i \in C_i}
| c = {{Defof|Cartesian Product}}
}}
{{eqn | r = \bigcap_{i \mathop \in I} \set {x \in S: x_i \in C_i}
}}
{{end-eqn}}
Thus by Intersection of Closed Sets is Closed in Topological Space,... | Let $\family {\struct {S_i, \tau_i} }_{i \mathop \in I}$ be a [[Definition:Indexed Family|family]] of [[Definition:Topological Space|topological spaces]] where $I$ is an arbitrary [[Definition:Indexing Set|indexing set]].
Let $\ds S = \prod_{i \mathop \in I} S_i$.
Let $T = \struct {S, \TT}$ be the [[Definition:Produc... | First note that:
{{begin-eqn}}
{{eqn | l = \prod_{i \mathop \in I} C_i
| r = \set {x \in S : \forall i \in I, x_i \in C_i}
| c = {{Defof|Cartesian Product}}
}}
{{eqn | r = \bigcap_{i \mathop \in I} \set {x \in S: x_i \in C_i}
}}
{{end-eqn}}
Thus by [[Intersection of Closed Sets is Closed in Topological S... | Product of Closed Sets is Closed | https://proofwiki.org/wiki/Product_of_Closed_Sets_is_Closed | https://proofwiki.org/wiki/Product_of_Closed_Sets_is_Closed | [
"Product Spaces",
"Closed Sets"
] | [
"Definition:Indexing Set/Family",
"Definition:Topological Space",
"Definition:Indexing Set",
"Definition:Product Space (Topology)",
"Definition:Product Topology",
"Definition:Indexing Set/Family",
"Definition:Closed Set/Topology",
"Definition:Closed Set/Topology"
] | [
"Intersection of Closed Sets is Closed/Topology",
"Definition:Closed Set/Topology",
"Definition:Closed Set/Topology",
"Definition:Open Set/Topology",
"Definition:Product Topology/Natural Sub-Basis",
"Definition:Open Set/Topology",
"Definition:Closed Set/Topology"
] |
proofwiki-13825 | Prime Number Formed by Concatenating Consecutive Integers down to 1 | Let $N$ be an integer whose decimal representation consists of the concatenation of all the integers from a given $n$ in descending order down to $1$.
Let the $N$ that is so formed be prime.
The only $n$ less than $100$ for which this is true is $82$.
That is:
:$82 \, 818 \, 079 \, 787 \, 776 \ldots 121 \, 110 \, 987 \... | Can be determined by checking all numbers formed in such a way for primality. | Let $N$ be an [[Definition:Integer|integer]] whose [[Definition:Decimal Notation|decimal representation]] consists of the concatenation of all the [[Definition:Integer|integers]] from a given $n$ in descending order down to $1$.
Let the $N$ that is so formed be [[Definition:Prime Number|prime]].
The only $n$ less tha... | Can be determined by checking all numbers formed in such a way for primality. | Prime Number Formed by Concatenating Consecutive Integers down to 1 | https://proofwiki.org/wiki/Prime_Number_Formed_by_Concatenating_Consecutive_Integers_down_to_1 | https://proofwiki.org/wiki/Prime_Number_Formed_by_Concatenating_Consecutive_Integers_down_to_1 | [
"Prime Numbers"
] | [
"Definition:Integer",
"Definition:Decimal Notation",
"Definition:Integer",
"Definition:Prime Number",
"Definition:Prime Number"
] | [] |
proofwiki-13826 | Naturality of Yoneda Lemma for Covariant Functors | Let $\sqbrk {\mathbf C, \mathbf {Set} }$ be the covariant functor category.
Let $\mathbf C \times \sqbrk {\mathbf C, \mathbf {Set} }$ be the product category.
Let $\mathbf C \times \sqbrk {\mathbf C, \mathbf {Set} } \to \mathbf {Set}: \tuple {A, F} \mapsto \map {\operatorname {Nat} } {h^A, F}$ be the covariant functor ... | By the Bijection in Yoneda Lemma for Covariant Functors, $\Phi_{\tuple {A, F} }$ is a bijection for all $\tuple {A, F}$.
Let $\tuple {f, \xi}: \tuple {A, F} \to \tuple {B, G} $ be a morphism in $\mathbf C \times \sqbrk {\mathbf C, \mathbf {Set} }$.
To prove that $\Phi$ is a natural isomorphism, it remains to prove that... | Let $\sqbrk {\mathbf C, \mathbf {Set} }$ be the [[Definition:Covariant Functor Category|covariant functor category]].
Let $\mathbf C \times \sqbrk {\mathbf C, \mathbf {Set} }$ be the [[Definition:Product Category|product category]].
Let $\mathbf C \times \sqbrk {\mathbf C, \mathbf {Set} } \to \mathbf {Set}: \tuple {A... | By the [[Bijection in Yoneda Lemma for Covariant Functors]], $\Phi_{\tuple {A, F} }$ is a [[Definition:Bijection|bijection]] for all $\tuple {A, F}$.
Let $\tuple {f, \xi}: \tuple {A, F} \to \tuple {B, G} $ be a [[Definition:Morphism|morphism]] in $\mathbf C \times \sqbrk {\mathbf C, \mathbf {Set} }$.
To prove that $\... | Naturality of Yoneda Lemma for Covariant Functors | https://proofwiki.org/wiki/Naturality_of_Yoneda_Lemma_for_Covariant_Functors | https://proofwiki.org/wiki/Naturality_of_Yoneda_Lemma_for_Covariant_Functors | [
"Functors",
"Natural Transformations"
] | [
"Definition:Functor Category",
"Definition:Product Category",
"Definition:Functor/Covariant",
"Definition:Composition of Functors",
"Definition:Hom Bifunctor",
"Definition:Product of Functors",
"Definition:Yoneda Functor/Contravariant",
"Definition:Identity Functor",
"Definition:Functor Evaluation B... | [
"Bijection in Yoneda Lemma for Covariant Functors",
"Definition:Bijection",
"Definition:Morphism",
"Definition:Natural Isomorphism",
"Definition:Commutative Diagram"
] |
proofwiki-13827 | Alternating Even-Odd Digit Palindromic Prime | Let the notation $\paren {abc}_n$ be interpreted to mean $n$ consecutive repetitions of a string of digits $abc$ concatenated in the decimal representation of an integer.
The integer:
:$\paren {10987654321234567890}_{42} 1$
has the following properties:
: it is a palindromic prime with $841$ digits
: its digits are alt... | {{Alpertron-factorizer|date = $22$nd March $2022$|time = $0.4$ seconds}}
This number has $20 \times 42 + 1 = 841$ digits.
The remaining properties of this number is obvious by inspection.
{{qed}} | Let the notation $\paren {abc}_n$ be interpreted to mean $n$ consecutive repetitions of a [[Definition:String|string]] of [[Definition:Digit|digits]] $abc$ concatenated in the [[Definition:Decimal Notation|decimal representation]] of an [[Definition:Integer|integer]].
The [[Definition:Integer|integer]]:
:$\paren {10... | {{Alpertron-factorizer|date = $22$nd March $2022$|time = $0.4$ seconds}}
This number has $20 \times 42 + 1 = 841$ [[Definition:Digit|digits]].
The remaining properties of this number is obvious by inspection.
{{qed}} | Alternating Even-Odd Digit Palindromic Prime | https://proofwiki.org/wiki/Alternating_Even-Odd_Digit_Palindromic_Prime | https://proofwiki.org/wiki/Alternating_Even-Odd_Digit_Palindromic_Prime | [
"Prime Numbers"
] | [
"Definition:String",
"Definition:Digit",
"Definition:Decimal Notation",
"Definition:Integer",
"Definition:Integer",
"Definition:Palindromic Prime",
"Definition:Digit",
"Definition:Digit",
"Definition:Odd Integer",
"Definition:Even Integer"
] | [
"Definition:Digit"
] |
proofwiki-13828 | Smallest Titanic Palindromic Prime | The smallest titanic prime that is also palindromic is:
:$10^{1000} + 81 \, 918 \times 10^{498} + 1$
which can be written as:
:$1 \underbrace {000 \ldots 000}_{497} 81918 \underbrace {000 \ldots 000}_{497} 1$ | {{Alpertron-factorizer|date = $6$th March $2022$|time = $1.7$ seconds}}
It remains to be demonstrated that it is the smallest such palindromic prime with $1000$ digits or more.
By 11 is Only Palindromic Prime with Even Number of Digits, there are no palindromic primes with exactly $1000$ digits.
Hence such a prime must... | The smallest [[Definition:Titanic Prime|titanic prime]] that is also [[Definition:Palindromic Prime|palindromic]] is:
:$10^{1000} + 81 \, 918 \times 10^{498} + 1$
which can be written as:
:$1 \underbrace {000 \ldots 000}_{497} 81918 \underbrace {000 \ldots 000}_{497} 1$ | {{Alpertron-factorizer|date = $6$th March $2022$|time = $1.7$ seconds}}
It remains to be demonstrated that it is the smallest such [[Definition:Palindromic Prime|palindromic prime]] with $1000$ [[Definition:Digit|digits]] or more.
By [[11 is Only Palindromic Prime with Even Number of Digits]], there are no [[Definit... | Smallest Titanic Palindromic Prime | https://proofwiki.org/wiki/Smallest_Titanic_Palindromic_Prime | https://proofwiki.org/wiki/Smallest_Titanic_Palindromic_Prime | [
"Prime Numbers"
] | [
"Definition:Titanic Prime",
"Definition:Palindromic Prime"
] | [
"Definition:Palindromic Prime",
"Definition:Digit",
"11 is Only Palindromic Prime with Even Number of Digits",
"Definition:Palindromic Prime",
"Definition:Digit",
"Definition:Prime Number",
"Definition:Prime Number",
"Definition:Titanic Prime",
"Definition:Palindromic Prime"
] |
proofwiki-13829 | Yoneda Embedding Theorem | Let $\mathbf C$ be a locally small category.
Let $\mathbf {Set}$ be the category of sets.
Let $\sqbrk { {\mathbf C}^{\operatorname {op} }, \mathbf {Set} }$ be the contravariant functor category.
Then the Yoneda embedding $h_- : {\mathbf C} \to \sqbrk { {\mathbf C}^{\operatorname {op} }, \mathbf {Set} }$ is a fully fait... | Let $X, Y \in \mathbf C$ be objects.
Then $\map {\operatorname {Hom} _{\mathbf C} } {X, Y}$ is a set, since $\mathbf C$ is locally small.
We need to show that $f \mapsto h_f$ induces a bjection:
:$\map {\operatorname {Hom} _{\mathbf C} } {X, Y} \to \map {\operatorname {Hom} _{\sqbrk { {\mathbf C}^{\operatorname {op} },... | Let $\mathbf C$ be a [[Definition:Locally Small Category|locally small category]].
Let $\mathbf {Set}$ be the [[Definition:Category of Sets|category of sets]].
Let $\sqbrk { {\mathbf C}^{\operatorname {op} }, \mathbf {Set} }$ be the [[Definition:Contravariant Functor Category|contravariant functor category]].
Then ... | Let $X, Y \in \mathbf C$ be [[Definition:Object (Category Theory)|objects]].
Then $\map {\operatorname {Hom} _{\mathbf C} } {X, Y}$ is a [[Definition:Set|set]], since $\mathbf C$ is [[Definition:Locally Small Category|locally small]].
We need to show that $f \mapsto h_f$ induces a [[Definition:Bijection|bjection]]:
... | Yoneda Embedding Theorem | https://proofwiki.org/wiki/Yoneda_Embedding_Theorem | https://proofwiki.org/wiki/Yoneda_Embedding_Theorem | [
"Category Theory"
] | [
"Definition:Locally Small Category",
"Definition:Category of Sets",
"Definition:Functor Category",
"Definition:Yoneda Functor/Yoneda Embedding",
"Definition:Fully Faithful Functor",
"Definition:Embedding of Categories"
] | [
"Definition:Object (Category Theory)",
"Definition:Set",
"Definition:Locally Small Category",
"Definition:Bijection",
"Yoneda Lemma for Contravariant Functors",
"Category:Category Theory"
] |
proofwiki-13830 | Bijection in Yoneda Lemma for Covariant Functors | Let $F: \mathbf C \to \mathbf {Set}$ be a covariant functor.
Let $A \in \mathbf C$ be an object.
Let $\operatorname{id}_A$ be its identity morphism.
Let $h^A = \map {\operatorname {Hom} } {A, -}$ be its covariant hom-functor.
The class of natural transformations $\map {\operatorname {Nat} } {h^A, F}$ is a small class, ... | The fact that $\map {\operatorname {Nat} } {h^A, F}$ is a small class follows when we prove that the mappings are bijections. | Let $F: \mathbf C \to \mathbf {Set}$ be a [[Definition:Covariant Functor|covariant functor]].
Let $A \in \mathbf C$ be an [[Definition:Object of Category|object]].
Let $\operatorname{id}_A$ be its [[Definition:Identity Morphism|identity morphism]].
Let $h^A = \map {\operatorname {Hom} } {A, -}$ be its [[Definition:C... | The fact that $\map {\operatorname {Nat} } {h^A, F}$ is a [[Definition:Small Class|small class]] follows when we prove that the mappings are bijections. | Bijection in Yoneda Lemma for Covariant Functors | https://proofwiki.org/wiki/Bijection_in_Yoneda_Lemma_for_Covariant_Functors | https://proofwiki.org/wiki/Bijection_in_Yoneda_Lemma_for_Covariant_Functors | [
"Functors",
"Natural Transformations"
] | [
"Definition:Functor/Covariant",
"Definition:Object (Category Theory)",
"Definition:Identity Morphism",
"Definition:Covariant Hom Functor",
"Definition:Class of Natural Transformations",
"Definition:Small Class",
"Definition:Inverse Mapping"
] | [
"Definition:Small Class"
] |
proofwiki-13831 | Naturality of Yoneda Lemma for Contravariant Functors | Let $[{\mathbf C}^{\operatorname{op}}, \mathbf{Set}]$ be the contravariant functor category.
Let ${\mathbf C}^{\operatorname{op}} \times [{\mathbf C}^{\operatorname{op}}, \mathbf{Set}] $ be the product category.
Let ${\mathbf C}^{\operatorname{op}} \times [{\mathbf C}^{\operatorname{op}}, \mathbf{Set}] \to \mathbf{Set}... | {{proof wanted|from Naturality of Yoneda Lemma for Covariant Functors, by duality}} | Let $[{\mathbf C}^{\operatorname{op}}, \mathbf{Set}]$ be the [[Definition:Contravariant Functor Category|contravariant functor category]].
Let ${\mathbf C}^{\operatorname{op}} \times [{\mathbf C}^{\operatorname{op}}, \mathbf{Set}] $ be the [[Definition:Product Category|product category]].
Let ${\mathbf C}^{\operatorn... | {{proof wanted|from [[Naturality of Yoneda Lemma for Covariant Functors]], by duality}} | Naturality of Yoneda Lemma for Contravariant Functors | https://proofwiki.org/wiki/Naturality_of_Yoneda_Lemma_for_Contravariant_Functors | https://proofwiki.org/wiki/Naturality_of_Yoneda_Lemma_for_Contravariant_Functors | [
"Functors",
"Natural Transformations"
] | [
"Definition:Functor Category",
"Definition:Product Category",
"Definition:Functor/Covariant",
"Definition:Composition of Functors",
"Definition:Hom Bifunctor",
"Definition:Product of Functors",
"Definition:Opposite Functor",
"Definition:Yoneda Functor/Yoneda Embedding",
"Definition:Identity Functor"... | [
"Naturality of Yoneda Lemma for Covariant Functors"
] |
proofwiki-13832 | Period of Reciprocal of Repunit 1031 is 1031 | The decimal expansion of the reciprocal of the repunit prime $R_{1031}$ has a period of $1031$.
:$\dfrac 1 {R_{1031}} = 0 \cdotp \underbrace{\dot 000 \ldots 000}_{1030} \dot 9$
{{refactor|Extract the fact that this is a unique period prime into another page}}
This is the only prime number to have a period of exactly $1... | The reciprocal of a repunit $R_n$ is of the form:
:$\dfrac 1 {R_n} = 0 \cdotp \underbrace{\dot 000 \ldots 000}_{n - 1} \dot 9$
Thus $\dfrac 1 {R_{1031}}$ has a period of $1031$.
From Period of Reciprocal of Prime, for prime numbers such that:
:$p \nmid 10$
we have that the period of such a prime is the order of $10$ mo... | The [[Definition:Decimal Expansion|decimal expansion]] of the [[Definition:Reciprocal|reciprocal]] of the [[Definition:Repunit Prime|repunit prime]] $R_{1031}$ has a [[Definition:Period of Recurrence|period]] of $1031$.
:$\dfrac 1 {R_{1031}} = 0 \cdotp \underbrace{\dot 000 \ldots 000}_{1030} \dot 9$
{{refactor|Extract... | The [[Definition:Reciprocal|reciprocal]] of a [[Definition:Repunit|repunit]] $R_n$ is of the form:
:$\dfrac 1 {R_n} = 0 \cdotp \underbrace{\dot 000 \ldots 000}_{n - 1} \dot 9$
Thus $\dfrac 1 {R_{1031}}$ has a [[Definition:Period of Recurrence|period]] of $1031$.
From [[Period of Reciprocal of Prime]], for [[Definiti... | Period of Reciprocal of Repunit 1031 is 1031 | https://proofwiki.org/wiki/Period_of_Reciprocal_of_Repunit_1031_is_1031 | https://proofwiki.org/wiki/Period_of_Reciprocal_of_Repunit_1031_is_1031 | [
"1031",
"Examples of Reciprocals"
] | [
"Definition:Decimal Expansion",
"Definition:Reciprocal",
"Definition:Repunit Prime",
"Definition:Basis Expansion/Recurrence/Period",
"Definition:Prime Number",
"Definition:Basis Expansion/Recurrence/Period"
] | [
"Definition:Reciprocal",
"Definition:Repunit",
"Definition:Basis Expansion/Recurrence/Period",
"Period of Reciprocal of Prime",
"Definition:Prime Number",
"Definition:Basis Expansion/Recurrence/Period",
"Definition:Prime Number",
"Definition:Multiplicative Order of Integer",
"Definition:Integer",
... |
proofwiki-13833 | Titanic Prime whose Digits are all Prime | The integer defined as:
:$7352 \times \dfrac {10^{1104} - 1} {10^4 - 1} + 1$
is a titanic prime all of whose digits are themselves prime.
That is:
:$\underbrace{7352}_{275} 7353$ | It is clear that the digits are instances of $7$, $3$, $5$ and $2$, and so are all prime.
It is also noted that it has $275 \times 4 + 4 = 1104$ digits, making it titanic.
{{Alpertron-factorizer|date = $6$th March $2022$|time = $2.1$ seconds}} | The [[Definition:Integer|integer]] defined as:
:$7352 \times \dfrac {10^{1104} - 1} {10^4 - 1} + 1$
is a [[Definition:Titanic Prime|titanic prime]] all of whose [[Definition:Digit|digits]] are themselves [[Definition:Prime Number|prime]].
That is:
:$\underbrace{7352}_{275} 7353$ | It is clear that the [[Definition:Digit|digits]] are instances of $7$, $3$, $5$ and $2$, and so are all [[Definition:Prime Number|prime]].
It is also noted that it has $275 \times 4 + 4 = 1104$ [[Definition:Digit|digits]], making it [[Definition:Titanic Prime|titanic]].
{{Alpertron-factorizer|date = $6$th March $202... | Titanic Prime whose Digits are all Prime | https://proofwiki.org/wiki/Titanic_Prime_whose_Digits_are_all_Prime | https://proofwiki.org/wiki/Titanic_Prime_whose_Digits_are_all_Prime | [
"Titanic Primes"
] | [
"Definition:Integer",
"Definition:Titanic Prime",
"Definition:Digit",
"Definition:Prime Number"
] | [
"Definition:Digit",
"Definition:Prime Number",
"Definition:Digit",
"Definition:Titanic Prime"
] |
proofwiki-13834 | Titanic Prime whose Digits are all 0 or 1 | The integer defined as:
:$10^{641} \times \dfrac {10^{640} - 1} 9 + 1$
is a titanic prime all of whose digits are either $0$ or $1$.
That is:
:$\paren 1_{640} \paren 0_{640} 1$
where $\paren a_b$ means $b$ instances of $a$ in a string. | It is clear that the digits are instances of $0$ and $1$.
It is also noted that it has $640 \times 2 + 1 = 1281$ digits, making it titanic.
{{Alpertron-factorizer|date = $6$th March $2022$|time = $2.5$ seconds}} | The [[Definition:Integer|integer]] defined as:
:$10^{641} \times \dfrac {10^{640} - 1} 9 + 1$
is a [[Definition:Titanic Prime|titanic prime]] all of whose [[Definition:Digit|digits]] are either $0$ or $1$.
That is:
:$\paren 1_{640} \paren 0_{640} 1$
where $\paren a_b$ means $b$ instances of $a$ in a string. | It is clear that the [[Definition:Digit|digits]] are instances of $0$ and $1$.
It is also noted that it has $640 \times 2 + 1 = 1281$ [[Definition:Digit|digits]], making it [[Definition:Titanic Prime|titanic]].
{{Alpertron-factorizer|date = $6$th March $2022$|time = $2.5$ seconds}} | Titanic Prime whose Digits are all 0 or 1 | https://proofwiki.org/wiki/Titanic_Prime_whose_Digits_are_all_0_or_1 | https://proofwiki.org/wiki/Titanic_Prime_whose_Digits_are_all_0_or_1 | [
"Titanic Primes"
] | [
"Definition:Integer",
"Definition:Titanic Prime",
"Definition:Digit"
] | [
"Definition:Digit",
"Definition:Digit",
"Definition:Titanic Prime"
] |
proofwiki-13835 | Titanic Sophie Germain Prime | The integer defined as:
:$39 \, 051 \times 2^{6001} - 1$
is a titanic prime which is also a Sophie Germain prime:
{{begin-eqn}}
{{eqn | o =
| r = 11820 \, 50794 \, 19125 \, 52383 \, 74423 \, 53078 \, 56017 \, 05024 \, 84819 \, 01689
}}
{{eqn | o =
| r = 74975 \, 95139 \, 68621 \, 89553 \, 48654 \, 81137 \... | At $1812$ digits, it is clear by definition that this prime is titanic.
{{Alpertron-factorizer|date = $6$th March $2022$|time = $4.7$ seconds}}
To show that it is in fact a Sophie Germain prime, we also need to check that:
:$2 \paren {39 \, 051 \times 2^{6001} - 1} + 1 = 39 \, 051 \times 2^{6002} - 1$
is also prime.
{{... | The [[Definition:Integer|integer]] defined as:
:$39 \, 051 \times 2^{6001} - 1$
is a [[Definition:Titanic Prime|titanic prime]] which is also a [[Definition:Sophie Germain Prime|Sophie Germain prime]]:
{{begin-eqn}}
{{eqn | o =
| r = 11820 \, 50794 \, 19125 \, 52383 \, 74423 \, 53078 \, 56017 \, 05024 \, 8481... | At $1812$ [[Definition:Digit|digits]], it is clear by definition that this prime is [[Definition:Titanic Prime|titanic]].
{{Alpertron-factorizer|date = $6$th March $2022$|time = $4.7$ seconds}}
To show that it is in fact a [[Definition:Sophie Germain Prime|Sophie Germain prime]], we also need to check that:
:$2 \par... | Titanic Sophie Germain Prime | https://proofwiki.org/wiki/Titanic_Sophie_Germain_Prime | https://proofwiki.org/wiki/Titanic_Sophie_Germain_Prime | [
"Titanic Primes",
"Sophie Germain Primes"
] | [
"Definition:Integer",
"Definition:Titanic Prime",
"Definition:Sophie Germain Prime"
] | [
"Definition:Digit",
"Definition:Titanic Prime",
"Definition:Sophie Germain Prime",
"Definition:Prime Number"
] |
proofwiki-13836 | Titanic Prime whose Digits are all 9 except for one 1 | The integer defined as:
:$2 \times 10^{3020} - 1$
is a titanic prime all of whose digits are $9$ except one, which is $1$.
That is:
:$1 \left({9}\right)_{3020}$
where $\left({a}\right)_b$ means $b$ instances of $a$ in a string. | It is clear that the digits are instances of $9$ except for the first $1$.
It is also noted that it has $3020 + 1 = 3021$ digits, making it titanic.
{{Alpertron-factorizer|date = $6$th March $2022$|time = $25.8$ seconds}} | The [[Definition:Integer|integer]] defined as:
:$2 \times 10^{3020} - 1$
is a [[Definition:Titanic Prime|titanic prime]] all of whose [[Definition:Digit|digits]] are $9$ except one, which is $1$.
That is:
:$1 \left({9}\right)_{3020}$
where $\left({a}\right)_b$ means $b$ instances of $a$ in a string. | It is clear that the [[Definition:Digit|digits]] are instances of $9$ except for the first $1$.
It is also noted that it has $3020 + 1 = 3021$ [[Definition:Digit|digits]], making it [[Definition:Titanic Prime|titanic]].
{{Alpertron-factorizer|date = $6$th March $2022$|time = $25.8$ seconds}} | Titanic Prime whose Digits are all 9 except for one 1 | https://proofwiki.org/wiki/Titanic_Prime_whose_Digits_are_all_9_except_for_one_1 | https://proofwiki.org/wiki/Titanic_Prime_whose_Digits_are_all_9_except_for_one_1 | [
"Titanic Primes"
] | [
"Definition:Integer",
"Definition:Titanic Prime",
"Definition:Digit"
] | [
"Definition:Digit",
"Definition:Digit",
"Definition:Titanic Prime"
] |
proofwiki-13837 | Titanic Prime consisting of 111 Blocks of each Digit plus Zeroes | The integer defined as:
:$\paren {123456789}_{111} \paren 0_{2284} 1$
where $\paren a_b$ means $b$ instances of $a$ in a string, is a titanic prime. | It is noted that it has $9 \times 111 + 2284 + 1 = 3284$ digits, making it titanic.
It can be expressed arithmetically as:
:$123456789 \times \dfrac {10^{999} - 1} {10^9 - 1} \times 10^{2285} + 1$
{{Alpertron-factorizer|date = $6$th March $2022$|time = $50.2$ seconds}} | The [[Definition:Integer|integer]] defined as:
:$\paren {123456789}_{111} \paren 0_{2284} 1$
where $\paren a_b$ means $b$ instances of $a$ in a string, is a [[Definition:Titanic Prime|titanic prime]]. | It is noted that it has $9 \times 111 + 2284 + 1 = 3284$ [[Definition:Digit|digits]], making it [[Definition:Titanic Prime|titanic]].
It can be expressed arithmetically as:
:$123456789 \times \dfrac {10^{999} - 1} {10^9 - 1} \times 10^{2285} + 1$
{{Alpertron-factorizer|date = $6$th March $2022$|time = $50.2$ seconds... | Titanic Prime consisting of 111 Blocks of each Digit plus Zeroes | https://proofwiki.org/wiki/Titanic_Prime_consisting_of_111_Blocks_of_each_Digit_plus_Zeroes | https://proofwiki.org/wiki/Titanic_Prime_consisting_of_111_Blocks_of_each_Digit_plus_Zeroes | [
"Titanic Primes"
] | [
"Definition:Integer",
"Definition:Titanic Prime"
] | [
"Definition:Digit",
"Definition:Titanic Prime"
] |
proofwiki-13838 | Titanic Prime whose Digits are all Odd | The integer defined as:
:$1358 \times 10^{3821} - 1$
is a titanic prime all of whose digits are odd.
That is:
:$1357 \paren 9_{3821}$
where $\paren a_b$ means $b$ instances of $a$ in a string. | It is clear that the digits are all instances of $9$ except for the initial $1357$, all of which are odd.
It is also noted that it has $4 + 3821 = 3825$ digits, making it titanic.
{{Alpertron-factorizer|date = $6$th March $2022$|time = $34.4$ seconds}} | The [[Definition:Integer|integer]] defined as:
:$1358 \times 10^{3821} - 1$
is a [[Definition:Titanic Prime|titanic prime]] all of whose [[Definition:Digit|digits]] are [[Definition:Odd Integer|odd]].
That is:
:$1357 \paren 9_{3821}$
where $\paren a_b$ means $b$ instances of $a$ in a string. | It is clear that the [[Definition:Digit|digits]] are all instances of $9$ except for the initial $1357$, all of which are [[Definition:Odd Integer|odd]].
It is also noted that it has $4 + 3821 = 3825$ [[Definition:Digit|digits]], making it [[Definition:Titanic Prime|titanic]].
{{Alpertron-factorizer|date = $6$th Mar... | Titanic Prime whose Digits are all Odd | https://proofwiki.org/wiki/Titanic_Prime_whose_Digits_are_all_Odd | https://proofwiki.org/wiki/Titanic_Prime_whose_Digits_are_all_Odd | [
"Titanic Primes"
] | [
"Definition:Integer",
"Definition:Titanic Prime",
"Definition:Digit",
"Definition:Odd Integer"
] | [
"Definition:Digit",
"Definition:Odd Integer",
"Definition:Digit",
"Definition:Titanic Prime"
] |
proofwiki-13839 | Pair of Titanic Twin Primes | The integers defined as:
:$190 \, 116 \times 3003 \times 10^{5120} \pm 1$
are a pair of titanic twin primes.
That is:
:$570 \, 918 \, 347 \paren 9_{5820}$
and:
:$570 \, 918 \, 348 \paren 0_{5819} 1$
where $\paren a_b$ means $b$ instances of $a$ in a string. | It is noted that these integers have $9 + 5820 = 5829$ digits, making them titanic.
{{Alpertron-factorizer|date = $6$th March $2022$|time = $45$ seconds}} | The [[Definition:Integer|integers]] defined as:
:$190 \, 116 \times 3003 \times 10^{5120} \pm 1$
are a pair of [[Definition:Titanic Prime|titanic]] [[Definition:Twin Primes|twin primes]].
That is:
:$570 \, 918 \, 347 \paren 9_{5820}$
and:
:$570 \, 918 \, 348 \paren 0_{5819} 1$
where $\paren a_b$ means $b$ instanc... | It is noted that these [[Definition:Integer|integers]] have $9 + 5820 = 5829$ [[Definition:Digit|digits]], making them [[Definition:Titanic Prime|titanic]].
{{Alpertron-factorizer|date = $6$th March $2022$|time = $45$ seconds}} | Pair of Titanic Twin Primes | https://proofwiki.org/wiki/Pair_of_Titanic_Twin_Primes | https://proofwiki.org/wiki/Pair_of_Titanic_Twin_Primes | [
"Titanic Primes",
"Twin Primes/Examples"
] | [
"Definition:Integer",
"Definition:Titanic Prime",
"Definition:Twin Primes"
] | [
"Definition:Integer",
"Definition:Digit",
"Definition:Titanic Prime"
] |
proofwiki-13840 | Gigantic Palindromic Prime | The integer defined as:
:$10^{11 \, 810} + 1 \, 465 \, 641 \times 10^{5902} + 1$
is a gigantic prime which is also palindromic.
That is:
:$1(0)_{5901}1465641(0)_{5901}1$
where $\left({a}\right)_b$ means $b$ instances of $a$ in a string. | It is clear that this number is palindromic.
It is also noted that it has $1 + 5901 + 7 + 5901 + 1 = 11 \, 811$ digits, making it gigantic.
{{Alpertron-factorizer|date = $6$th March $2022$|time = $4$ minutes}} | The [[Definition:Integer|integer]] defined as:
:$10^{11 \, 810} + 1 \, 465 \, 641 \times 10^{5902} + 1$
is a [[Definition:Gigantic Prime|gigantic prime]] which is also [[Definition:Palindromic Prime|palindromic]].
That is:
:$1(0)_{5901}1465641(0)_{5901}1$
where $\left({a}\right)_b$ means $b$ instances of $a$ in a ... | It is clear that this number is [[Definition:Palindromic Prime|palindromic]].
It is also noted that it has $1 + 5901 + 7 + 5901 + 1 = 11 \, 811$ [[Definition:Digit|digits]], making it [[Definition:Gigantic Prime|gigantic]].
{{Alpertron-factorizer|date = $6$th March $2022$|time = $4$ minutes}} | Gigantic Palindromic Prime | https://proofwiki.org/wiki/Gigantic_Palindromic_Prime | https://proofwiki.org/wiki/Gigantic_Palindromic_Prime | [
"Gigantic Primes",
"Palindromic Primes"
] | [
"Definition:Integer",
"Definition:Gigantic Prime",
"Definition:Palindromic Prime"
] | [
"Definition:Palindromic Prime",
"Definition:Digit",
"Definition:Gigantic Prime"
] |
proofwiki-13841 | Zeroth Hyperoperation is Successor Function | The '''zeroth hyperoperation''' is the successor function:
:$H_0 \left({x, y}\right) = y + 1$ | Immediate by definition of the successor function $s: \Z_{\ge 0} \to \Z_{\ge 0}$:
:$\forall y \in \Z_{\ge 0}: s \left({y}\right) = y + 1$
and for the $n$th hyperoperation:
$\forall n, x, y \in \Z_{\ge 0}: H_n \left({x, y}\right) = \begin{cases}
y + 1 & : n = 0 \\
x & : n = 1, y = 0 \\
0 & : n = 2, y = 0 \\
1 & : n > 2... | The '''[[Definition:Nth Hyperoperation|zeroth hyperoperation]]''' is the [[Definition:Successor Mapping|successor function]]:
:$H_0 \left({x, y}\right) = y + 1$ | Immediate by definition of the [[Definition:Successor Mapping|successor function]] $s: \Z_{\ge 0} \to \Z_{\ge 0}$:
:$\forall y \in \Z_{\ge 0}: s \left({y}\right) = y + 1$
and for the [[Definition:Nth Hyperoperation|$n$th hyperoperation]]:
$\forall n, x, y \in \Z_{\ge 0}: H_n \left({x, y}\right) = \begin{cases}
y + 1... | Zeroth Hyperoperation is Successor Function | https://proofwiki.org/wiki/Zeroth_Hyperoperation_is_Successor_Function | https://proofwiki.org/wiki/Zeroth_Hyperoperation_is_Successor_Function | [
"Hyperoperation"
] | [
"Definition:Hyperoperation/Nth Hyperoperation",
"Definition:Successor Mapping"
] | [
"Definition:Successor Mapping",
"Definition:Hyperoperation/Nth Hyperoperation",
"Definition:Hyperoperation/Nth Hyperoperation",
"Definition:Degenerate Case",
"Definition:Mapping",
"Definition:Operation/Operand",
"Category:Hyperoperation"
] |
proofwiki-13842 | First Hyperoperation is Addition Operation | The '''$1$st hyperoperation''' is the addition operation restricted to the positive integers:
:$\forall x, y \in \Z_{\ge 0}: H_1 \left({x, y}\right) = x + y$ | By definition of the hyperoperation sequence:
:$\forall n, x, y \in \Z_{\ge 0}: H_n \left({x, y}\right) = \begin{cases}
y + 1 & : n = 0 \\
x & : n = 1, y = 0 \\
0 & : n = 2, y = 0 \\
1 & : n > 2, y = 0 \\
H_{n - 1} \left({x, H_n \left({x, y - 1}\right)}\right) & : n > 0, y > 0 \end{cases}$
Thus the $1$st hyperoperation... | The '''[[Definition:Nth Hyperoperation|$1$st hyperoperation]]''' is the [[Definition:Integer Addition|addition operation]] [[Definition:Restriction of Operation|restricted]] to the [[Definition:Positive Integer|positive integers]]:
:$\forall x, y \in \Z_{\ge 0}: H_1 \left({x, y}\right) = x + y$ | By definition of the [[Definition:Hyperoperation Sequence|hyperoperation sequence]]:
:$\forall n, x, y \in \Z_{\ge 0}: H_n \left({x, y}\right) = \begin{cases}
y + 1 & : n = 0 \\
x & : n = 1, y = 0 \\
0 & : n = 2, y = 0 \\
1 & : n > 2, y = 0 \\
H_{n - 1} \left({x, H_n \left({x, y - 1}\right)}\right) & : n > 0, y > 0 \e... | First Hyperoperation is Addition Operation | https://proofwiki.org/wiki/First_Hyperoperation_is_Addition_Operation | https://proofwiki.org/wiki/First_Hyperoperation_is_Addition_Operation | [
"Hyperoperation"
] | [
"Definition:Hyperoperation/Nth Hyperoperation",
"Definition:Addition/Integers",
"Definition:Restriction/Operation",
"Definition:Positive/Integer"
] | [
"Definition:Hyperoperation/Sequence",
"Definition:Hyperoperation/Nth Hyperoperation",
"Zeroth Hyperoperation is Successor Function",
"Principle of Mathematical Induction",
"Definition:Proposition",
"Principle of Mathematical Induction",
"Principle of Mathematical Induction",
"Principle of Mathematical... |
proofwiki-13843 | Second Hyperoperation is Multiplication Operation | The '''$2$nd hyperoperation''' is the multiplication operation restricted to the positive integers:
:$\forall x, y \in \Z_{\ge 0}: H_2 \left({x, y}\right) = x \times y$ | By definition of the hyperoperation sequence:
:$\forall n, x, y \in \Z_{\ge 0}: H_n \left({x, y}\right) = \begin{cases}
y + 1 & : n = 0 \\
x & : n = 1, y = 0 \\
0 & : n = 2, y = 0 \\
1 & : n > 2, y = 0 \\
H_{n - 1} \left({x, H_n \left({x, y - 1}\right)}\right) & : n > 0, y > 0 \end{cases}$
Thus the $2$nd hyperoperation... | The '''[[Definition:Nth Hyperoperation|$2$nd hyperoperation]]''' is the [[Definition:Integer Multiplication|multiplication operation]] [[Definition:Restriction of Operation|restricted]] to the [[Definition:Positive Integer|positive integers]]:
:$\forall x, y \in \Z_{\ge 0}: H_2 \left({x, y}\right) = x \times y$ | By definition of the [[Definition:Hyperoperation Sequence|hyperoperation sequence]]:
:$\forall n, x, y \in \Z_{\ge 0}: H_n \left({x, y}\right) = \begin{cases}
y + 1 & : n = 0 \\
x & : n = 1, y = 0 \\
0 & : n = 2, y = 0 \\
1 & : n > 2, y = 0 \\
H_{n - 1} \left({x, H_n \left({x, y - 1}\right)}\right) & : n > 0, y > 0 \e... | Second Hyperoperation is Multiplication Operation | https://proofwiki.org/wiki/Second_Hyperoperation_is_Multiplication_Operation | https://proofwiki.org/wiki/Second_Hyperoperation_is_Multiplication_Operation | [
"Hyperoperation"
] | [
"Definition:Hyperoperation/Nth Hyperoperation",
"Definition:Multiplication/Integers",
"Definition:Restriction/Operation",
"Definition:Positive/Integer"
] | [
"Definition:Hyperoperation/Sequence",
"Definition:Hyperoperation/Nth Hyperoperation",
"First Hyperoperation is Addition Operation",
"Principle of Mathematical Induction",
"Definition:Proposition",
"Principle of Mathematical Induction",
"Principle of Mathematical Induction",
"Principle of Mathematical ... |
proofwiki-13844 | Third Hyperoperation is Integer Power Operation | The '''$3$rd hyperoperation''' is the integer power operation restricted to the positive integers:
:$\forall x, y \in \Z_{\ge 0}: H_3 \left({x, y}\right) = x^y$ | By definition of the hyperoperation sequence:
:$\forall n, x, y \in \Z_{\ge 0}: H_n \left({x, y}\right) = \begin{cases}
y + 1 & : n = 0 \\
x & : n = 1, y = 0 \\
0 & : n = 2, y = 0 \\
1 & : n > 2, y = 0 \\
H_{n - 1} \left({x, H_n \left({x, y - 1}\right)}\right) & : n > 0, y > 0 \end{cases}$
Thus the $3$rd hyperoperation... | The '''[[Definition:Nth Hyperoperation|$3$rd hyperoperation]]''' is the [[Definition:Integer Power|integer power operation]] [[Definition:Restriction of Operation|restricted]] to the [[Definition:Positive Integer|positive integers]]:
:$\forall x, y \in \Z_{\ge 0}: H_3 \left({x, y}\right) = x^y$ | By definition of the [[Definition:Hyperoperation Sequence|hyperoperation sequence]]:
:$\forall n, x, y \in \Z_{\ge 0}: H_n \left({x, y}\right) = \begin{cases}
y + 1 & : n = 0 \\
x & : n = 1, y = 0 \\
0 & : n = 2, y = 0 \\
1 & : n > 2, y = 0 \\
H_{n - 1} \left({x, H_n \left({x, y - 1}\right)}\right) & : n > 0, y > 0 \e... | Third Hyperoperation is Integer Power Operation | https://proofwiki.org/wiki/Third_Hyperoperation_is_Integer_Power_Operation | https://proofwiki.org/wiki/Third_Hyperoperation_is_Integer_Power_Operation | [
"Hyperoperation"
] | [
"Definition:Hyperoperation/Nth Hyperoperation",
"Definition:Power (Algebra)/Integer",
"Definition:Restriction/Operation",
"Definition:Positive/Integer"
] | [
"Definition:Hyperoperation/Sequence",
"Definition:Hyperoperation/Nth Hyperoperation",
"Second Hyperoperation is Multiplication Operation",
"Principle of Mathematical Induction",
"Definition:Proposition",
"Principle of Mathematical Induction",
"Principle of Mathematical Induction",
"Principle of Mathem... |
proofwiki-13845 | Fourth Hyperoperation is Tetration Operation | The '''$4$th hyperoperation''' is the tetration operation restricted to the positive integers:
:$\forall x, y \in \Z_{\ge 0}: H_4 \left({x, y}\right) = x \uparrow \uparrow y$
where $\uparrow \uparrow$ denotes tetration:
:$x \uparrow \uparrow n := \begin{cases}
1 & : n = 0 \\
x \uparrow \left({x \uparrow \uparrow \left(... | By definition of the hyperoperation sequence:
:$\forall n, x, y \in \Z_{\ge 0}: H_n \left({x, y}\right) = \begin{cases}
y + 1 & : n = 0 \\
x & : n = 1, y = 0 \\
0 & : n = 2, y = 0 \\
1 & : n > 2, y = 0 \\
H_{n - 1} \left({x, H_n \left({x, y - 1}\right)}\right) & : n > 0, y > 0 \end{cases}$
Thus the $4$th hyperoperation... | The '''[[Definition:Nth Hyperoperation|$4$th hyperoperation]]''' is the [[Definition:Tetration|tetration operation]] [[Definition:Restriction of Operation|restricted]] to the [[Definition:Positive Integer|positive integers]]:
:$\forall x, y \in \Z_{\ge 0}: H_4 \left({x, y}\right) = x \uparrow \uparrow y$
where $\upar... | By definition of the [[Definition:Hyperoperation Sequence|hyperoperation sequence]]:
:$\forall n, x, y \in \Z_{\ge 0}: H_n \left({x, y}\right) = \begin{cases}
y + 1 & : n = 0 \\
x & : n = 1, y = 0 \\
0 & : n = 2, y = 0 \\
1 & : n > 2, y = 0 \\
H_{n - 1} \left({x, H_n \left({x, y - 1}\right)}\right) & : n > 0, y > 0 \e... | Fourth Hyperoperation is Tetration Operation | https://proofwiki.org/wiki/Fourth_Hyperoperation_is_Tetration_Operation | https://proofwiki.org/wiki/Fourth_Hyperoperation_is_Tetration_Operation | [
"Hyperoperation"
] | [
"Definition:Hyperoperation/Nth Hyperoperation",
"Definition:Tetration",
"Definition:Restriction/Operation",
"Definition:Positive/Integer",
"Definition:Tetration",
"Definition:Power (Algebra)/Integer/Knuth Notation",
"Definition:Power (Algebra)/Integer"
] | [
"Definition:Hyperoperation/Sequence",
"Definition:Hyperoperation/Nth Hyperoperation",
"Third Hyperoperation is Integer Power Operation",
"Principle of Mathematical Induction",
"Definition:Proposition",
"Principle of Mathematical Induction",
"Principle of Mathematical Induction",
"Principle of Mathemat... |
proofwiki-13846 | Largest Integer Expressible by 3 Digits/Logarithm Base 10 | :$\map {\log_{10} } {9^{9^9} } \approx 369 \, 693 \,099 \cdotp 63157 \, 03685 \, 87876 \, 1$ | {{begin-eqn}}
{{eqn | l = \map {\log_{10} } {9^{9^9} }
| r = 9^9 \times \log_{10} 9
| c =
}}
{{eqn | o = \approx
| r = 387 \, 420 \, 489 \times 0\cdotp 95424 \, 25094 \, 393249
| c = by calculator
}}
{{eqn | o = \approx
| r = 369 \, 693 \, 099 \cdotp 63157 \, 03685 \, 87876 \, 1
| c... | :$\map {\log_{10} } {9^{9^9} } \approx 369 \, 693 \,099 \cdotp 63157 \, 03685 \, 87876 \, 1$ | {{begin-eqn}}
{{eqn | l = \map {\log_{10} } {9^{9^9} }
| r = 9^9 \times \log_{10} 9
| c =
}}
{{eqn | o = \approx
| r = 387 \, 420 \, 489 \times 0\cdotp 95424 \, 25094 \, 393249
| c = by calculator
}}
{{eqn | o = \approx
| r = 369 \, 693 \, 099 \cdotp 63157 \, 03685 \, 87876 \, 1
| c... | Largest Integer Expressible by 3 Digits/Logarithm Base 10 | https://proofwiki.org/wiki/Largest_Integer_Expressible_by_3_Digits/Logarithm_Base_10 | https://proofwiki.org/wiki/Largest_Integer_Expressible_by_3_Digits/Logarithm_Base_10 | [
"Largest Integer Expressible by 3 Digits"
] | [] | [] |
proofwiki-13847 | Largest Integer Expressible by 3 Digits/Number of Digits | :$9^{9^9}$ has $369 \, 693 \, 100$ digits when expressed in decimal notation. | Let $n$ be the number of digits in $9^{9^9}$
From Number of Digits in Number:
:$n = 1 + \floor {\map {\log_{10} } {9^{9^9} } }$
where $\floor {\ldots}$ denotes the floor function.
Then:
{{begin-eqn}}
{{eqn | l = \map {\log_{10} } {9^{9^9} }
| o = \approx
| r = 369 \, 693 \, 099 \cdotp 63157 \, 03685 \, 8787... | :$9^{9^9}$ has $369 \, 693 \, 100$ [[Definition:Digit|digits]] when expressed in [[Definition:Decimal Notation|decimal notation]]. | Let $n$ be the number of [[Definition:Digit|digits]] in $9^{9^9}$
From [[Number of Digits in Number]]:
:$n = 1 + \floor {\map {\log_{10} } {9^{9^9} } }$
where $\floor {\ldots}$ denotes the [[Definition:Floor Function|floor function]].
Then:
{{begin-eqn}}
{{eqn | l = \map {\log_{10} } {9^{9^9} }
| o = \approx
... | Largest Integer Expressible by 3 Digits/Number of Digits | https://proofwiki.org/wiki/Largest_Integer_Expressible_by_3_Digits/Number_of_Digits | https://proofwiki.org/wiki/Largest_Integer_Expressible_by_3_Digits/Number_of_Digits | [
"Largest Integer Expressible by 3 Digits"
] | [
"Definition:Digit",
"Definition:Decimal Notation"
] | [
"Definition:Digit",
"Number of Digits in Number",
"Definition:Floor Function",
"Largest Integer Expressible by 3 Digits/Logarithm Base 10"
] |
proofwiki-13848 | Compact Subset of Compact Space is not necessarily Closed | A compact subset of a compact space is not necessarily closed. | Let $S$ be a set containing more than one element.
Let $\tau = \set {S, \O}$ be the indiscrete topology on $S$.
Let $x \in S$.
Let $H = S \setminus \set x$.
Then $H$ is a proper subset of $S$.
Then from Subset of Indiscrete Space is Compact and Sequentially Compact, the subspace induced by $\tau$ on $H$ is a compact su... | A [[Definition:Compact Topological Subspace|compact]] [[Definition:Subset|subset]] of a [[Definition:Compact Topological Space|compact space]] is not necessarily [[Definition:Closed Set (Topology)|closed]]. | Let $S$ be a [[Definition:Set|set]] containing more than one [[Definition:Element|element]].
Let $\tau = \set {S, \O}$ be the [[Definition:Indiscrete Topology|indiscrete topology]] on $S$.
Let $x \in S$.
Let $H = S \setminus \set x$.
Then $H$ is a [[Definition:Proper Subset|proper subset]] of $S$.
Then from [[Subs... | Compact Subset of Compact Space is not necessarily Closed | https://proofwiki.org/wiki/Compact_Subset_of_Compact_Space_is_not_necessarily_Closed | https://proofwiki.org/wiki/Compact_Subset_of_Compact_Space_is_not_necessarily_Closed | [
"Closed Sets",
"Compact Topological Spaces"
] | [
"Definition:Compact Topological Space/Subspace",
"Definition:Subset",
"Definition:Compact Topological Space",
"Definition:Closed Set/Topology"
] | [
"Definition:Set",
"Definition:Element",
"Definition:Indiscrete Topology",
"Definition:Proper Subset",
"Subset of Indiscrete Space is Compact and Sequentially Compact",
"Definition:Topological Subspace",
"Definition:Compact Topological Space/Subspace",
"Closed Sets in Indiscrete Topology",
"Definitio... |
proofwiki-13849 | Separability is not Weakly Hereditary | The property of separability is not weakly hereditary. | It needs to be demonstrated that there exists a separable topological space which has a subspace which is closed but not separable.
Consider an uncountable particular point space $T = \struct {S, \tau_p}$.
From Particular Point Space is Separable, $T$ is separable.
By definition, the particular point $p$ is an open poi... | The property of [[Definition:Separable Space|separability]] is not [[Definition:Weakly Hereditary Property|weakly hereditary]]. | It needs to be demonstrated that there exists a [[Definition:Separable Space|separable topological space]] which has a [[Definition:Topological Subspace|subspace]] which is [[Definition:Closed Set (Topology)|closed]] but not [[Definition:Separable Space|separable]].
Consider an [[Definition:Uncountable Particular Poi... | Separability is not Weakly Hereditary | https://proofwiki.org/wiki/Separability_is_not_Weakly_Hereditary | https://proofwiki.org/wiki/Separability_is_not_Weakly_Hereditary | [
"Separable Spaces",
"Examples of Weakly Hereditary Properties"
] | [
"Definition:Separable Space",
"Definition:Weakly Hereditary Property"
] | [
"Definition:Separable Space",
"Definition:Topological Subspace",
"Definition:Closed Set/Topology",
"Definition:Separable Space",
"Definition:Particular Point Topology/Uncountable",
"Particular Point Space is Separable",
"Definition:Separable Space",
"Definition:Particular Point Topology",
"Definitio... |
proofwiki-13850 | Filter is Finer iff Sets of Basis are Subsets | Let $S$ be a set.
Let $\powerset S$ denote the power set of $S$.
Let $\FF, \FF' \subset \powerset S$ be two filters on $S$.
Let $\FF$ have a basis $\BB$.
Let $\FF'$ have a basis $\BB'$.
$\FF$ is finer than $\FF'$ {{iff}} for every set of $\BB'$, there is a set of $\BB$ subset to it. | === Necessary Condition ===
Suppose for every set of $\BB'$, there is a set of $\BB$ subset to it.
Pick any $U \in \FF'$.
Then from definition of a basis:
:$\exists V' \in \BB': V' \subseteq U$
By our assumption:
:$\exists V \in \BB: V \subseteq V' \subseteq U \subseteq S$
By definition of a filter:
:$U \in \FF$.
Hence... | Let $S$ be a [[Definition:Set|set]].
Let $\powerset S$ denote the [[Definition:Power Set|power set]] of $S$.
Let $\FF, \FF' \subset \powerset S$ be two [[Definition:Filter on Set|filters]] on $S$.
Let $\FF$ have a [[Definition:Filter Basis|basis]] $\BB$.
Let $\FF'$ have a [[Definition:Filter Basis|basis]] $\BB'$.
... | === Necessary Condition ===
Suppose for every [[Definition:Set|set]] of $\BB'$, there is a [[Definition:Set|set]] of $\BB$ [[Definition:Subset|subset]] to it.
Pick any $U \in \FF'$.
Then from definition of a [[Definition:Filter Basis|basis]]:
:$\exists V' \in \BB': V' \subseteq U$
By our assumption:
:$\exists V \in... | Filter is Finer iff Sets of Basis are Subsets | https://proofwiki.org/wiki/Filter_is_Finer_iff_Sets_of_Basis_are_Subsets | https://proofwiki.org/wiki/Filter_is_Finer_iff_Sets_of_Basis_are_Subsets | [
"Finer Filters on Sets",
"Filter Bases"
] | [
"Definition:Set",
"Definition:Power Set",
"Definition:Filter on Set",
"Definition:Filter Basis",
"Definition:Filter Basis",
"Definition:Finer Filter on Set",
"Definition:Set",
"Definition:Set",
"Definition:Subset"
] | [
"Definition:Set",
"Definition:Set",
"Definition:Subset",
"Definition:Filter Basis",
"Definition:Filter on Set",
"Definition:Set",
"Definition:Set",
"Definition:Subset",
"Definition:Set",
"Definition:Filter Basis"
] |
proofwiki-13851 | Existence of Topological Space which satisfies no Separation Axioms | There exists at least one example of a topological space for which none of the Tychonoff separation axioms are satisfied. | ;Proof by Counterexample
Let $T$ be the topological space consisting of the double pointed topology on the countable complement topology on an uncountable set.
From Double Pointed Countable Complement Topology fulfils no Separation Axioms, we have that $T$ satisfies none of the Tychonoff separation axioms.
{{qed}} | There exists at least one example of a [[Definition:Topological Space|topological space]] for which none of the [[Definition:Tychonoff Separation Axioms|Tychonoff separation axioms]] are satisfied. | ;[[Proof by Counterexample]]
Let $T$ be the [[Definition:Topological Space|topological space]] consisting of the [[Definition:Double Pointed Topology|double pointed topology]] on the [[Definition:Countable Complement Topology|countable complement topology]] on an [[Definition:Uncountable Set|uncountable set]].
From [... | Existence of Topological Space which satisfies no Separation Axioms | https://proofwiki.org/wiki/Existence_of_Topological_Space_which_satisfies_no_Separation_Axioms | https://proofwiki.org/wiki/Existence_of_Topological_Space_which_satisfies_no_Separation_Axioms | [
"Separation Axioms"
] | [
"Definition:Topological Space",
"Definition:Tychonoff Separation Axioms"
] | [
"Proof by Counterexample",
"Definition:Topological Space",
"Definition:Double Pointed Topology",
"Definition:Countable Complement Topology",
"Definition:Uncountable/Set",
"Double Pointed Countable Complement Topology fulfils no Separation Axioms",
"Definition:Tychonoff Separation Axioms"
] |
proofwiki-13852 | T0 Space is not necessarily T1, T2, T3, T4 or T5 | There exists at least one example of a topological space for which none of the Tychonoff separation axioms are satisfied except for the $T_0$ (Kolmogorov) axiom. | Let $T$ be the overlapping interval space.
From Overlapping Interval Space fulfils no Separation Axioms but $T_0$, we have that $T$ satisfies none of the Tychonoff separation axioms except for the $T_0$ axiom.
{{qed}} | There exists at least one example of a [[Definition:Topological Space|topological space]] for which none of the [[Definition:Tychonoff Separation Axioms|Tychonoff separation axioms]] are satisfied except for the [[Definition:T0 Space|$T_0$ (Kolmogorov) axiom]]. | Let $T$ be the [[Definition:Overlapping Interval Topology|overlapping interval space]].
From [[Overlapping Interval Space fulfils no Separation Axioms but T0|Overlapping Interval Space fulfils no Separation Axioms but $T_0$]], we have that $T$ satisfies none of the [[Definition:Tychonoff Separation Axioms|Tychonoff se... | T0 Space is not necessarily T1, T2, T3, T4 or T5 | https://proofwiki.org/wiki/T0_Space_is_not_necessarily_T1,_T2,_T3,_T4_or_T5 | https://proofwiki.org/wiki/T0_Space_is_not_necessarily_T1,_T2,_T3,_T4_or_T5 | [
"T0 Spaces",
"Separation Axioms"
] | [
"Definition:Topological Space",
"Definition:Tychonoff Separation Axioms",
"Definition:T0 Space"
] | [
"Definition:Overlapping Interval Topology",
"Overlapping Interval Space fulfils no Separation Axioms but T0",
"Definition:Tychonoff Separation Axioms",
"Definition:T0 Space"
] |
proofwiki-13853 | T1 Space is not necessarily T2, T3, T4 or T5 | There exists at least one example of a topological space for which none of the Tychonoff separation axioms are satisfied except for the $T_0$ axiom and $T_1$ axiom. | ;Proof by Counterexample
Let $T$ be the finite complement topology on a countable space.
From Finite Complement Space is $T_1$, $T$ is a $T_1$ (Fréchet) space.
Hence from $T_1$ Space is $T_0$, also a $T_0$ (Kolmogorov) space.
The rest of the result follows from:
:Finite Complement Space is not $T_2$
and:
:Finite Comple... | There exists at least one example of a [[Definition:Topological Space|topological space]] for which none of the [[Definition:Tychonoff Separation Axioms|Tychonoff separation axioms]] are satisfied except for the [[Definition:T0 Space|$T_0$ axiom]] and [[Definition:T1 Space|$T_1$ axiom]]. | ;[[Proof by Counterexample]]
Let $T$ be the [[Definition:Countable Finite Complement Topology|finite complement topology on a countable space]].
From [[Finite Complement Space is T1|Finite Complement Space is $T_1$]], $T$ is a [[Definition:T1 Space|$T_1$ (Fréchet) space]].
Hence from [[T1 Space is T0|$T_1$ Space is... | T1 Space is not necessarily T2, T3, T4 or T5 | https://proofwiki.org/wiki/T1_Space_is_not_necessarily_T2,_T3,_T4_or_T5 | https://proofwiki.org/wiki/T1_Space_is_not_necessarily_T2,_T3,_T4_or_T5 | [
"Separation Axioms",
"T0 Spaces",
"T1 Spaces"
] | [
"Definition:Topological Space",
"Definition:Tychonoff Separation Axioms",
"Definition:T0 Space",
"Definition:T1 Space"
] | [
"Proof by Counterexample",
"Definition:Finite Complement Topology/Countable",
"Finite Complement Space is T1",
"Definition:T1 Space",
"T1 Space is T0",
"Definition:T0 Space",
"Finite Complement Space is not T2",
"Finite Complement Space is not T3",
"Finite Complement Space is not T4",
"Finite Comp... |
proofwiki-13854 | T2 Space is not necessarily T3, T4 or T5 | A topological space which is a $T_2$ (Hausdorff) space (and hence also a $T_0$ space and $T_1$ space) is not necessarily a $T_3$ space, a $T_4$ space or a $T_5$ space. | Let $T$ be an irrational slope topological space.
We have:
:Irrational Slope Space is $T_2$
:Irrational Slope Space is $T_1$
:Irrational Slope Space is $T_0$
Thus $T$ is a $T_2$ (Hausdorff) space, a $T_1$ space and a $T_0$ space.
The rest of the result follows from:
:Irrational Slope Space is not $T_3$
:Irrational Slop... | A [[Definition:Topological Space|topological space]] which is a [[Definition:T2 Space|$T_2$ (Hausdorff) space]] (and hence also a [[Definition:T0 Space|$T_0$ space]] and [[Definition:T1 Space|$T_1$ space]]) is not necessarily a [[Definition:T3 Space|$T_3$ space]], a [[Definition:T4 Space|$T_4$ space]] or a [[Definition... | Let $T$ be an [[Definition:Irrational Slope Topology|irrational slope topological space]].
We have:
:[[Irrational Slope Space is T2|Irrational Slope Space is $T_2$]]
:[[Irrational Slope Space is T1|Irrational Slope Space is $T_1$]]
:[[Irrational Slope Space is T0|Irrational Slope Space is $T_0$]]
Thus $T$ is a [[Def... | T2 Space is not necessarily T3, T4 or T5 | https://proofwiki.org/wiki/T2_Space_is_not_necessarily_T3,_T4_or_T5 | https://proofwiki.org/wiki/T2_Space_is_not_necessarily_T3,_T4_or_T5 | [
"Hausdorff Spaces",
"T3 Spaces",
"T4 Spaces",
"T5 Spaces",
"Separation Axioms"
] | [
"Definition:Topological Space",
"Definition:T2 Space",
"Definition:T0 Space",
"Definition:T1 Space",
"Definition:T3 Space",
"Definition:T4 Space",
"Definition:T5 Space"
] | [
"Definition:Irrational Slope Topology",
"Irrational Slope Space is T2",
"Irrational Slope Space is T1",
"Irrational Slope Space is T0",
"Definition:T2 Space",
"Definition:T1 Space",
"Definition:T0 Space",
"Irrational Slope Space is not T3",
"Irrational Slope Space is not T4",
"Irrational Slope Spa... |
proofwiki-13855 | T3 Space is not necessarily T0, T1, T2, T4 or T5 | There exists at least one example of a topological space for which none of the Tychonoff separation axioms are satisfied except for the $T_3$ axiom. | Proof by Counterexample:
Let $T$ be the topological space consisting of the double pointed topology on the Tychonoff corkscrew topology.
From Double Pointed Tychonoff Corkscrew fulfils no Separation Axioms but $T_3$, we have that $T$ satisfies none of the Tychonoff separation axioms except for the $T_3$ axiom.
{{qed}} | There exists at least one example of a [[Definition:Topological Space|topological space]] for which none of the [[Definition:Tychonoff Separation Axioms|Tychonoff separation axioms]] are satisfied except for the [[Definition:T3 Space|$T_3$ axiom]]. | [[Proof by Counterexample]]:
Let $T$ be the [[Definition:Topological Space|topological space]] consisting of the [[Definition:Double Pointed Topology|double pointed topology]] on the [[Definition:Tychonoff Corkscrew|Tychonoff corkscrew topology]].
From [[Double Pointed Tychonoff Corkscrew fulfils no Separation Axioms... | T3 Space is not necessarily T0, T1, T2, T4 or T5 | https://proofwiki.org/wiki/T3_Space_is_not_necessarily_T0,_T1,_T2,_T4_or_T5 | https://proofwiki.org/wiki/T3_Space_is_not_necessarily_T0,_T1,_T2,_T4_or_T5 | [
"T3 Spaces",
"Separation Axioms"
] | [
"Definition:Topological Space",
"Definition:Tychonoff Separation Axioms",
"Definition:T3 Space"
] | [
"Proof by Counterexample",
"Definition:Topological Space",
"Definition:Double Pointed Topology",
"Definition:Tychonoff Corkscrew",
"Double Pointed Tychonoff Corkscrew fulfils no Separation Axioms but T3",
"Definition:Tychonoff Separation Axioms",
"Definition:T3 Space"
] |
proofwiki-13856 | T4 Space is not necessarily T0, T1, T2, T3 or T5 | There exists at least one example of a topological space for which none of the Tychonoff separation axioms are satisfied except for the $T_4$ axiom. | Let $T_S = \struct {S, \tau}$ be a countable complement topology on an uncountable set $S$.
Let $D = \struct {\set {0, 1}, \vartheta}$ be the indiscrete topology on two points.
Let $T_S \times D$ be the double pointed topology on $T$.
Let $\paren {T_S \times D}^*_{\bar p}$ be the open extension topology on $S \times \s... | There exists at least one example of a [[Definition:Topological Space|topological space]] for which none of the [[Definition:Tychonoff Separation Axioms|Tychonoff separation axioms]] are satisfied except for the [[Definition:T4 Space|$T_4$ axiom]]. | Let $T_S = \struct {S, \tau}$ be a [[Definition:Countable Complement Topology|countable complement topology]] on an [[Definition:Uncountable Set|uncountable set]] $S$.
Let $D = \struct {\set {0, 1}, \vartheta}$ be the [[Definition:Indiscrete Topology|indiscrete topology]] on two points.
Let $T_S \times D$ be the [[De... | T4 Space is not necessarily T0, T1, T2, T3 or T5 | https://proofwiki.org/wiki/T4_Space_is_not_necessarily_T0,_T1,_T2,_T3_or_T5 | https://proofwiki.org/wiki/T4_Space_is_not_necessarily_T0,_T1,_T2,_T3_or_T5 | [
"T4 Spaces",
"Separation Axioms"
] | [
"Definition:Topological Space",
"Definition:Tychonoff Separation Axioms",
"Definition:T4 Space"
] | [
"Definition:Countable Complement Topology",
"Definition:Uncountable/Set",
"Definition:Indiscrete Topology",
"Definition:Double Pointed Topology",
"Definition:Open Extension Topology",
"Open Extension of Double Pointed Countable Complement Topology is T4",
"Definition:Tychonoff Separation Axioms",
"Def... |
proofwiki-13857 | Compact Hausdorff Space is not necessarily T5 | A compact $T_2$ (Hausdorff) space is not necessarily a $T_5$ space. | Let $T$ be the Tychonoff plank.
From Tychonoff Plank is Compact, $T$ is a compact topological space.
From Tychonoff Plank is $T_2$, $T$ is a $T_2$ (Hausdorff) space.
But from Tychonoff Plank is not $T_5$, $T$ is not a $T_5$ space.
{{qed}} | A [[Definition:Compact Topological Space|compact]] [[Definition:T2 Space|$T_2$ (Hausdorff) space]] is not necessarily a [[Definition:T5 Space|$T_5$ space]]. | Let $T$ be the [[Definition:Tychonoff Plank|Tychonoff plank]].
From [[Tychonoff Plank is Compact]], $T$ is a [[Definition:Compact Topological Space|compact topological space]].
From [[Tychonoff Plank is T2|Tychonoff Plank is $T_2$]], $T$ is a [[Definition:T2 Space|$T_2$ (Hausdorff) space]].
But from [[Tychonoff Pl... | Compact Hausdorff Space is not necessarily T5 | https://proofwiki.org/wiki/Compact_Hausdorff_Space_is_not_necessarily_T5 | https://proofwiki.org/wiki/Compact_Hausdorff_Space_is_not_necessarily_T5 | [
"Hausdorff Spaces",
"Compact Topological Spaces",
"T5 Spaces",
"Separation Axioms"
] | [
"Definition:Compact Topological Space",
"Definition:T2 Space",
"Definition:T5 Space"
] | [
"Definition:Tychonoff Plank",
"Tychonoff Plank is Compact",
"Definition:Compact Topological Space",
"Tychonoff Plank is T2",
"Definition:T2 Space",
"Tychonoff Plank is not T5",
"Definition:T5 Space"
] |
proofwiki-13858 | Regular Space is not necessarily Completely Regular | A topological space which is a regular space is not necessarily also a completely regular space. | Let $T$ be a Tychonoff corkscrew.
From Tychonoff Corkscrew is Regular, $T$ is a regular space.
From Tychonoff Corkscrew is not Completely Regular, $T$ is not a completely regular space.
Hence the result.
{{qed}} | A [[Definition:Topological Space|topological space]] which is a [[Definition:Regular Space|regular space]] is not necessarily also a [[Definition:Completely Regular Space|completely regular space]]. | Let $T$ be a [[Definition:Tychonoff Corkscrew|Tychonoff corkscrew]].
From [[Tychonoff Corkscrew is Regular]], $T$ is a [[Definition:Regular Space|regular space]].
From [[Tychonoff Corkscrew is not Completely Regular]], $T$ is not a [[Definition:Completely Regular Space|completely regular space]].
Hence the result.
... | Regular Space is not necessarily Completely Regular | https://proofwiki.org/wiki/Regular_Space_is_not_necessarily_Completely_Regular | https://proofwiki.org/wiki/Regular_Space_is_not_necessarily_Completely_Regular | [
"Regular Spaces",
"Completely Regular Spaces"
] | [
"Definition:Topological Space",
"Definition:Regular Space",
"Definition:Completely Regular Space"
] | [
"Definition:Tychonoff Corkscrew",
"Tychonoff Corkscrew is Regular",
"Definition:Regular Space",
"Tychonoff Corkscrew is not Completely Regular",
"Definition:Completely Regular Space"
] |
proofwiki-13859 | Completely Regular Space is not necessarily Normal | A topological space which is a completely regular space is not necessarily also a normal space. | Let $T$ be the Niemytzki plane.
From Niemytzki Plane is Completely Regular, $T$ is a completely regular space.
From Niemytzki Plane is not Normal, $T$ is not a normal space.
Hence the result.
{{qed}} | A [[Definition:Topological Space|topological space]] which is a [[Definition:Completely Regular Space|completely regular space]] is not necessarily also a [[Definition:Normal Space|normal space]]. | Let $T$ be the [[Definition:Niemytzki Plane|Niemytzki plane]].
From [[Niemytzki Plane is Completely Regular]], $T$ is a [[Definition:Completely Regular Space|completely regular space]].
From [[Niemytzki Plane is not Normal]], $T$ is not a [[Definition:Normal Space|normal space]].
Hence the result.
{{qed}} | Completely Regular Space is not necessarily Normal | https://proofwiki.org/wiki/Completely_Regular_Space_is_not_necessarily_Normal | https://proofwiki.org/wiki/Completely_Regular_Space_is_not_necessarily_Normal | [
"Normal Spaces",
"Completely Regular Spaces"
] | [
"Definition:Topological Space",
"Definition:Completely Regular Space",
"Definition:Normal Space"
] | [
"Definition:Niemytzki Plane",
"Niemytzki Plane is Completely Regular",
"Definition:Completely Regular Space",
"Niemytzki Plane is not Normal",
"Definition:Normal Space"
] |
proofwiki-13860 | T4 Space is not necessarily T3.5 | A topological space which is a $T_4$ space is not necessarily also a $T_{3 \frac 1 2}$ space. | Let $T$ be a Hjalmar Ekdal space.
From Hjalmar Ekdal Space is $T_4$, $T$ is a $T_4$ space.
From Hjalmar Ekdal Space is not $T_{3 \frac 1 2}$, $T$ is not a $T_{3 \frac 1 2}$ space.
Hence the result.
{{qed}} | A [[Definition:Topological Space|topological space]] which is a [[Definition:T4 Space|$T_4$ space]] is not necessarily also a [[Definition:T3.5 Space|$T_{3 \frac 1 2}$ space]]. | Let $T$ be a [[Definition:Hjalmar Ekdal Topology|Hjalmar Ekdal space]].
From [[Hjalmar Ekdal Space is T4|Hjalmar Ekdal Space is $T_4$]], $T$ is a [[Definition:T4 Space|$T_4$ space]].
From [[Hjalmar Ekdal Space is not T3.5|Hjalmar Ekdal Space is not $T_{3 \frac 1 2}$]], $T$ is not a [[Definition:T3.5 Space|$T_{3 \fra... | T4 Space is not necessarily T3.5 | https://proofwiki.org/wiki/T4_Space_is_not_necessarily_T3.5 | https://proofwiki.org/wiki/T4_Space_is_not_necessarily_T3.5 | [
"Separation Axioms",
"T4 Spaces",
"T3.5 Spaces"
] | [
"Definition:Topological Space",
"Definition:T4 Space",
"Definition:T3.5 Space"
] | [
"Definition:Hjalmar Ekdal Topology",
"Hjalmar Ekdal Space is T4",
"Definition:T4 Space",
"Hjalmar Ekdal Space is not T3.5",
"Definition:T3.5 Space"
] |
proofwiki-13861 | Normal Space is not necessarily Completely Normal | A topological space which is a normal space is not necessarily also a completely normal space. | Let $T$ be a Tychonoff plank.
From Tychonoff Plank is Normal, $T$ is a normal space.
From Tychonoff Plank is not Completely Normal, $T$ is not a completely normal space.
Hence the result.
{{qed}} | A [[Definition:Topological Space|topological space]] which is a [[Definition:Normal Space|normal space]] is not necessarily also a [[Definition:Completely Normal Space|completely normal space]]. | Let $T$ be a [[Definition:Tychonoff Plank|Tychonoff plank]].
From [[Tychonoff Plank is Normal]], $T$ is a [[Definition:Normal Space|normal space]].
From [[Tychonoff Plank is not Completely Normal]], $T$ is not a [[Definition:Completely Normal Space|completely normal space]].
Hence the result.
{{qed}} | Normal Space is not necessarily Completely Normal | https://proofwiki.org/wiki/Normal_Space_is_not_necessarily_Completely_Normal | https://proofwiki.org/wiki/Normal_Space_is_not_necessarily_Completely_Normal | [
"Normal Spaces",
"Completely Normal Spaces"
] | [
"Definition:Topological Space",
"Definition:Normal Space",
"Definition:Completely Normal Space"
] | [
"Definition:Tychonoff Plank",
"Tychonoff Plank is Normal",
"Definition:Normal Space",
"Tychonoff Plank is not Completely Normal",
"Definition:Completely Normal Space"
] |
proofwiki-13862 | T2.5 Space is not necessarily Regular | There exists at least one example of a topological space which is a $T_{2 \frac 1 2}$ space, but is not also a regular space. | Let $T$ be a half-disc space.
From Half-Disc Space is $T_{2 \frac 1 2}$, $T$ is a $T_{2 \frac 1 2}$ space.
From Half-Disc Space is not Regular, $T$ is not a regular space.
Hence the result.
{{qed}} | There exists at least one example of a [[Definition:Topological Space|topological space]] which is a [[Definition:T2.5 Space|$T_{2 \frac 1 2}$ space]], but is not also a [[Definition:Regular Space|regular space]]. | Let $T$ be a [[Definition:Half-Disc Topology|half-disc space]].
From [[Half-Disc Space is T2.5|Half-Disc Space is $T_{2 \frac 1 2}$]], $T$ is a [[Definition:T2.5 Space|$T_{2 \frac 1 2}$ space]].
From [[Half-Disc Space is not Regular]], $T$ is not a [[Definition:Regular Space|regular space]].
Hence the result.
{{qed... | T2.5 Space is not necessarily Regular | https://proofwiki.org/wiki/T2.5_Space_is_not_necessarily_Regular | https://proofwiki.org/wiki/T2.5_Space_is_not_necessarily_Regular | [
"T2.5 Spaces",
"Regular Spaces"
] | [
"Definition:Topological Space",
"Definition:T2.5 Space",
"Definition:Regular Space"
] | [
"Definition:Half-Disc Topology",
"Half-Disc Space is T2.5",
"Definition:T2.5 Space",
"Half-Disc Space is not Regular",
"Definition:Regular Space"
] |
proofwiki-13863 | T2 Space is not necessarily T2.5 | A topological space which is a $T_2$ (Hausdorff) space is not necessarily also a $T_{2 \frac 1 2}$ space. | Let $T$ be an irrational slope space.
From Irrational Slope Space is $T_2$, $T$ is a $T_2$ (Hausdorff) space.
From Irrational Slope Space is not $T_{2 \frac 1 2}$, $T$ is not a $T_{2 \frac 1 2}$ space.
{{qed}} | A [[Definition:Topological Space|topological space]] which is a [[Definition:T2 Space|$T_2$ (Hausdorff) space]] is not necessarily also a [[Definition:T2.5 Space|$T_{2 \frac 1 2}$ space]]. | Let $T$ be an [[Definition:Irrational Slope Space|irrational slope space]].
From [[Irrational Slope Space is T2|Irrational Slope Space is $T_2$]], $T$ is a [[Definition:T2 Space|$T_2$ (Hausdorff) space]].
From [[Irrational Slope Space is not T2.5|Irrational Slope Space is not $T_{2 \frac 1 2}$]], $T$ is not a [[Defi... | T2 Space is not necessarily T2.5 | https://proofwiki.org/wiki/T2_Space_is_not_necessarily_T2.5 | https://proofwiki.org/wiki/T2_Space_is_not_necessarily_T2.5 | [
"Hausdorff Spaces",
"T2.5 Spaces"
] | [
"Definition:Topological Space",
"Definition:T2 Space",
"Definition:T2.5 Space"
] | [
"Definition:Irrational Slope Topology",
"Irrational Slope Space is T2",
"Definition:T2 Space",
"Irrational Slope Space is not T2.5",
"Definition:T2.5 Space"
] |
proofwiki-13864 | Separation Properties Not Preserved under Expansion | Let $S$ be a set.
Let $T_A = \struct {S, \tau_1}$ and $T_B = \struct {S, \tau_2}$ be topological spaces such that $\tau_2$ is an expansion of $\tau_1$.
These separation properties are not generally preserved under expansion:
{{help|Can we determine whether the semiregular property is preserved under expansion? We know ... | Let $\struct {\R, \tau_1}$ be the set of real numbers under the usual (Euclidean) topology.
Let $\struct {\R, \tau_2}$ be the indiscrete rational extension of $\struct {\R, \tau_1}$.
By definition, $\struct {\R, \tau_2}$ is an expansion of $\struct {\R, \tau_1}$.
From Metric Space fulfils all Separation Axioms, $\struc... | Let $S$ be a [[Definition:Set|set]].
Let $T_A = \struct {S, \tau_1}$ and $T_B = \struct {S, \tau_2}$ be [[Definition:Topological Space|topological spaces]] such that $\tau_2$ is an [[Definition:Expansion of Topology|expansion]] of $\tau_1$.
These [[Definition:Separation Axioms|separation properties]] are not general... | Let $\struct {\R, \tau_1}$ be the [[Definition:Real Number|set of real numbers]] under the [[Definition:Euclidean Topology on Real Number Line|usual (Euclidean) topology]].
Let $\struct {\R, \tau_2}$ be the [[Definition:Indiscrete Rational Extension of Reals|indiscrete rational extension]] of $\struct {\R, \tau_1}$.
... | Separation Properties Not Preserved under Expansion | https://proofwiki.org/wiki/Separation_Properties_Not_Preserved_under_Expansion | https://proofwiki.org/wiki/Separation_Properties_Not_Preserved_under_Expansion | [
"Separation Properties Not Preserved under Expansion",
"Separation Axioms",
"Expansions of Topologies"
] | [
"Definition:Set",
"Definition:Topological Space",
"Definition:Expansion of Topology",
"Definition:Tychonoff Separation Axioms",
"Definition:Expansion of Topology",
"Definition:Semiregular Space",
"T3 Property is Not Preserved under Expansion",
"Regular Property is Not Preserved under Expansion",
"T3... | [
"Definition:Real Number",
"Definition:Euclidean Space/Euclidean Topology/Real Number Line",
"Definition:Indiscrete Extension of Reals/Rational",
"Definition:Expansion of Topology",
"Metric Space fulfils all Separation Axioms",
"Definition:T3 Space",
"Definition:Regular Space",
"Definition:T3.5 Space",... |
proofwiki-13865 | Existence of Completely Normal Space whose Product Space is Not Normal | There exists at least one example of a completely normal topological space $T$ such that the product space $T \times T$ is not a normal topological space. | Let $T$ be the Sorgenfrey line.
Let $T' = T \times T$ be Sorgenfrey's half-open square.
From Sorgenfrey Line satisfies all Separation Axioms, $T$ is a completely normal space.
From Sorgenfrey's Half-Open Square is not Normal, $T'$ is not a normal space.
Hence the result.
{{qed}} | There exists at least one example of a [[Definition:Completely Normal Space|completely normal topological space]] $T$ such that the [[Definition:Product Space (Topology) of Two Factor Spaces|product space]] $T \times T$ is not a [[Definition:Normal Space|normal topological space]]. | Let $T$ be the [[Definition:Sorgenfrey Line|Sorgenfrey line]].
Let $T' = T \times T$ be [[Definition:Sorgenfrey's Half-Open Square|Sorgenfrey's half-open square]].
From [[Sorgenfrey Line satisfies all Separation Axioms]], $T$ is a [[Definition:Completely Normal Space|completely normal space]].
From [[Sorgenfrey's Ha... | Existence of Completely Normal Space whose Product Space is Not Normal | https://proofwiki.org/wiki/Existence_of_Completely_Normal_Space_whose_Product_Space_is_Not_Normal | https://proofwiki.org/wiki/Existence_of_Completely_Normal_Space_whose_Product_Space_is_Not_Normal | [
"Normal Spaces",
"Completely Normal Spaces",
"Product Spaces"
] | [
"Definition:Completely Normal Space",
"Definition:Product Space (Topology)/Two Factor Spaces",
"Definition:Normal Space"
] | [
"Definition:Sorgenfrey Line",
"Definition:Sorgenfrey's Half-Open Square Topology",
"Sorgenfrey Line satisfies all Separation Axioms",
"Definition:Completely Normal Space",
"Sorgenfrey's Half-Open Square is not Normal",
"Definition:Normal Space"
] |
proofwiki-13866 | Completely Normal Space is not necessarily Perfectly Normal | A completely normal topological space is not necessarily perfectly normal. | Let $T$ be an uncountable Fort space.
From Fort Space is Completely Normal, $T$ is a completely normal space.
From Uncountable Fort Space is not Perfectly Normal, $T$ is not a perfectly normal space.
Hence the result.
{{qed}} | A [[Definition:Completely Normal Space|completely normal topological space]] is not necessarily [[Definition:Perfectly Normal Space|perfectly normal]]. | Let $T$ be an [[Definition:Uncountable Fort Space|uncountable Fort space]].
From [[Fort Space is Completely Normal]], $T$ is a [[Definition:Completely Normal Space|completely normal space]].
From [[Uncountable Fort Space is not Perfectly Normal]], $T$ is not a [[Definition:Perfectly Normal Space|perfectly normal spa... | Completely Normal Space is not necessarily Perfectly Normal | https://proofwiki.org/wiki/Completely_Normal_Space_is_not_necessarily_Perfectly_Normal | https://proofwiki.org/wiki/Completely_Normal_Space_is_not_necessarily_Perfectly_Normal | [
"Completely Normal Spaces",
"Perfectly Normal Spaces"
] | [
"Definition:Completely Normal Space",
"Definition:Perfectly Normal Space"
] | [
"Definition:Fort Space/Uncountable",
"Fort Space is Completely Normal",
"Definition:Completely Normal Space",
"Uncountable Fort Space is not Perfectly Normal",
"Definition:Perfectly Normal Space"
] |
proofwiki-13867 | Semiregular Topological Space is not necessarily T3 | A semiregular topological space is not necessarily a $T_3$ space. | Let $T$ be a simplified Arens square.
From Simplified Arens Square is Semiregular:
:$T$ is a semiregular space.
From Simplified Arens Square is not $T_3$:
:$T$ is not a $T_3$ space.
Hence the result.
{{qed}} | A [[Definition:Semiregular Space|semiregular topological space]] is not necessarily a [[Definition:T3 Space|$T_3$ space]]. | Let $T$ be a [[Definition:Simplified Arens Square|simplified Arens square]].
From [[Simplified Arens Square is Semiregular]]:
:$T$ is a [[Definition:Semiregular Space|semiregular space]].
From [[Simplified Arens Square is not T3|Simplified Arens Square is not $T_3$]]:
:$T$ is not a [[Definition:T3 Space|$T_3$ space]]... | Semiregular Topological Space is not necessarily T3 | https://proofwiki.org/wiki/Semiregular_Topological_Space_is_not_necessarily_T3 | https://proofwiki.org/wiki/Semiregular_Topological_Space_is_not_necessarily_T3 | [
"T3 Spaces",
"Semiregular Spaces"
] | [
"Definition:Semiregular Space",
"Definition:T3 Space"
] | [
"Definition:Simplified Arens Square",
"Simplified Arens Square is Semiregular",
"Definition:Semiregular Space",
"Simplified Arens Square is not T3",
"Definition:T3 Space"
] |
proofwiki-13868 | Semiregular Topological Space is not necessarily T2.5 | A semiregular topological space is not necessarily a $T_{2 \frac 1 2}$ space. | Let $T$ be a simplified Arens square.
From Simplified Arens Square is Semiregular:
:$T$ is a semiregular space.
From Simplified Arens Square is not $T_{2 \frac 1 2}$:
:$T$ is not a $T_{2 \frac 1 2}$ space.
Hence the result.
{{qed}} | A [[Definition:Semiregular Space|semiregular topological space]] is not necessarily a [[Definition:T2.5 Space|$T_{2 \frac 1 2}$ space]]. | Let $T$ be a [[Definition:Simplified Arens Square|simplified Arens square]].
From [[Simplified Arens Square is Semiregular]]:
:$T$ is a [[Definition:Semiregular Space|semiregular space]].
From [[Simplified Arens Square is not T2.5|Simplified Arens Square is not $T_{2 \frac 1 2}$]]:
:$T$ is not a [[Definition:T2.5 Spa... | Semiregular Topological Space is not necessarily T2.5 | https://proofwiki.org/wiki/Semiregular_Topological_Space_is_not_necessarily_T2.5 | https://proofwiki.org/wiki/Semiregular_Topological_Space_is_not_necessarily_T2.5 | [
"T2.5 Spaces",
"Semiregular Spaces"
] | [
"Definition:Semiregular Space",
"Definition:T2.5 Space"
] | [
"Definition:Simplified Arens Square",
"Simplified Arens Square is Semiregular",
"Definition:Semiregular Space",
"Simplified Arens Square is not T2.5",
"Definition:T2.5 Space"
] |
proofwiki-13869 | Semiregular Topological Space is not necessarily Urysohn | A semiregular topological space is not necessarily a Urysohn space. | Let $T$ be an Arens square.
From Arens Square is Semiregular, $T$ is a semiregular space.
From Arens Square is not Urysohn, $T$ is not a Urysohn space.
Hence the result.
{{qed}} | A [[Definition:Semiregular Space|semiregular topological space]] is not necessarily a [[Definition:Urysohn Space|Urysohn space]]. | Let $T$ be an [[Definition:Arens Square|Arens square]].
From [[Arens Square is Semiregular]], $T$ is a [[Definition:Semiregular Space|semiregular space]].
From [[Arens Square is not Urysohn]], $T$ is not a [[Definition:Urysohn Space|Urysohn space]].
Hence the result.
{{qed}} | Semiregular Topological Space is not necessarily Urysohn | https://proofwiki.org/wiki/Semiregular_Topological_Space_is_not_necessarily_Urysohn | https://proofwiki.org/wiki/Semiregular_Topological_Space_is_not_necessarily_Urysohn | [
"Semiregular Spaces",
"Urysohn Spaces"
] | [
"Definition:Semiregular Space",
"Definition:Urysohn Space"
] | [
"Definition:Arens Square",
"Arens Square is Semiregular",
"Definition:Semiregular Space",
"Arens Square is not Urysohn",
"Definition:Urysohn Space"
] |
proofwiki-13870 | Topological Space is Compact iff Every Filter has Adherent Point | A topological space $T = \struct {S, \tau}$ is '''compact''' {{iff}} every filter on $S$ has an adherent point in $S$. | {{Recall|Compact Topological Space|compact topological space|index = 2}}
{{:Definition:Compact Topological Space/Definition 2}} | A [[Definition:Topological Space|topological space]] $T = \struct {S, \tau}$ is '''[[Definition:Compact Topological Space|compact]]''' {{iff}} every [[Definition:Filter on Set|filter]] on $S$ has an [[Definition:Adherent Point of Filter|adherent point]] in $S$. | {{Recall|Compact Topological Space|compact topological space|index = 2}}
{{:Definition:Compact Topological Space/Definition 2}} | Topological Space is Compact iff Every Filter has Adherent Point | https://proofwiki.org/wiki/Topological_Space_is_Compact_iff_Every_Filter_has_Adherent_Point | https://proofwiki.org/wiki/Topological_Space_is_Compact_iff_Every_Filter_has_Adherent_Point | [
"Topological Space is Compact iff Every Filter has Adherent Point",
"Compact Topological Spaces",
"Filters on Sets",
"Adherent Points"
] | [
"Definition:Topological Space",
"Definition:Compact Topological Space",
"Definition:Filter on Set",
"Definition:Adherent Point/Filter"
] | [] |
proofwiki-13871 | Sigma-Compact Space is not necessarily Compact | A $\sigma$-compact topological space is not necessarily also a compact space. | Let $T = \struct {\R, \tau_d}$ be the real number line with the usual (Euclidean) topology.
From Real Number Line is $\sigma$-Compact, $T$ is a $\sigma$-compact space.
From Real Number Line is not Countably Compact, $T$ is not a countably compact space.
The result follows from Compact Space is Countably Compact.
{{qed}... | A [[Definition:Sigma-Compact Space|$\sigma$-compact topological space]] is not necessarily also a [[Definition:Compact Topological Space|compact space]]. | Let $T = \struct {\R, \tau_d}$ be the [[Definition:Real Number Line with Euclidean Topology|real number line with the usual (Euclidean) topology]].
From [[Real Number Line is Sigma-Compact|Real Number Line is $\sigma$-Compact]], $T$ is a [[Definition:Sigma-Compact Space|$\sigma$-compact space]].
From [[Real Number L... | Sigma-Compact Space is not necessarily Compact | https://proofwiki.org/wiki/Sigma-Compact_Space_is_not_necessarily_Compact | https://proofwiki.org/wiki/Sigma-Compact_Space_is_not_necessarily_Compact | [
"Sigma-Compact Spaces",
"Compact Topological Spaces",
"Sequence of Implications of Global Compactness Properties"
] | [
"Definition:Sigma-Compact Space",
"Definition:Compact Topological Space"
] | [
"Definition:Euclidean Space/Euclidean Topology/Real Number Line",
"Real Number Line is Sigma-Compact",
"Definition:Sigma-Compact Space",
"Real Number Line is not Countably Compact",
"Definition:Countably Compact Space",
"Compact Space is Countably Compact"
] |
proofwiki-13872 | Lindelöf Space is not necessarily Sigma-Compact | A Lindelöf space is not necessarily also a compact space. | Let $T$ be the Sorgenfrey line.
From Sorgenfrey Line is Lindelöf, $T$ is a Lindelöf space.
From Sorgenfrey Line is not $\sigma$-Compact, $T$ is not a $\sigma$-compact space.
Hence the result.
{{qed}} | A [[Definition:Lindelöf Space|Lindelöf space]] is not necessarily also a [[Definition:Compact Topological Space|compact space]]. | Let $T$ be the [[Definition:Sorgenfrey Line|Sorgenfrey line]].
From [[Sorgenfrey Line is Lindelöf]], $T$ is a [[Definition:Lindelöf Space|Lindelöf space]].
From [[Sorgenfrey Line is not Sigma-Compact|Sorgenfrey Line is not $\sigma$-Compact]], $T$ is not a [[Definition:Sigma-Compact Space|$\sigma$-compact space]].
H... | Lindelöf Space is not necessarily Sigma-Compact | https://proofwiki.org/wiki/Lindelöf_Space_is_not_necessarily_Sigma-Compact | https://proofwiki.org/wiki/Lindelöf_Space_is_not_necessarily_Sigma-Compact | [
"Lindelöf Spaces",
"Sigma-Compact Spaces",
"Sequence of Implications of Global Compactness Properties"
] | [
"Definition:Lindelöf Space",
"Definition:Compact Topological Space"
] | [
"Definition:Sorgenfrey Line",
"Sorgenfrey Line is Lindelöf",
"Definition:Lindelöf Space",
"Sorgenfrey Line is not Sigma-Compact",
"Definition:Sigma-Compact Space"
] |
proofwiki-13873 | Compact Space is not necessarily Sequentially Compact | There exists at least one example of a compact topological space which is not also a sequentially compact space. | Let $T = \Bbb I^\Bbb I$ be the uncountable Cartesian product of the closed unit interval under the usual (Euclidean) topology.
From Uncountable Cartesian Product of Closed Unit Interval is Compact Space, $T$ is a compact space.
From Uncountable Cartesian Product of Closed Unit Interval is not Sequentially Compact Space... | There exists at least one example of a [[Definition:Compact Topological Space|compact topological space]] which is not also a [[Definition:Sequentially Compact Space|sequentially compact space]]. | Let $T = \Bbb I^\Bbb I$ be the [[Definition:Uncountable Cartesian Product of Closed Unit Interval|uncountable Cartesian product of the closed unit interval]] under the [[Definition:Euclidean Topology on Real Number Line|usual (Euclidean) topology]].
From [[Uncountable Cartesian Product of Closed Unit Interval is Comp... | Compact Space is not necessarily Sequentially Compact | https://proofwiki.org/wiki/Compact_Space_is_not_necessarily_Sequentially_Compact | https://proofwiki.org/wiki/Compact_Space_is_not_necessarily_Sequentially_Compact | [
"Sequentially Compact Spaces",
"Compact Topological Spaces",
"Sequence of Implications of Global Compactness Properties"
] | [
"Definition:Compact Topological Space",
"Definition:Sequentially Compact Space"
] | [
"Definition:Uncountable Cartesian Product of Closed Unit Interval",
"Definition:Euclidean Space/Euclidean Topology/Real Number Line",
"Uncountable Cartesian Product of Closed Unit Interval is Compact Space",
"Definition:Compact Topological Space",
"Uncountable Cartesian Product of Closed Unit Interval is no... |
proofwiki-13874 | Compact Space is not necessarily Sequentially Compact | There exists at least one example of a compact topological space which is not also a sequentially compact space. | Let $T := \hointr 0 \Omega \times \Bbb I^\Bbb I$ be the Cartesian product of $\hointr 0 \Omega$ under the interval topology, with the uncountable Cartesian product of the closed unit interval under the usual (Euclidean) topology.
From Cartesian Product of Open Ordinal Space with Uncountable Cartesian Product of Closed ... | There exists at least one example of a [[Definition:Compact Topological Space|compact topological space]] which is not also a [[Definition:Sequentially Compact Space|sequentially compact space]]. | Let $T := \hointr 0 \Omega \times \Bbb I^\Bbb I$ be the [[Definition:Cartesian Product|Cartesian product]] of $\hointr 0 \Omega$ under the [[Definition:Interval Topology|interval topology]], with the [[Definition:Uncountable Set|uncountable]] [[Definition:Cartesian Product|Cartesian product]] of the [[Definition:Closed... | Countably Compact Space is not necessarily Sequentially Compact/Proof 1 | https://proofwiki.org/wiki/Compact_Space_is_not_necessarily_Sequentially_Compact | https://proofwiki.org/wiki/Countably_Compact_Space_is_not_necessarily_Sequentially_Compact/Proof_1 | [
"Sequentially Compact Spaces",
"Compact Topological Spaces",
"Sequence of Implications of Global Compactness Properties"
] | [
"Definition:Compact Topological Space",
"Definition:Sequentially Compact Space"
] | [
"Definition:Cartesian Product",
"Definition:Order Topology",
"Definition:Uncountable/Set",
"Definition:Cartesian Product",
"Definition:Real Interval/Unit Interval/Closed",
"Definition:Euclidean Space/Euclidean Topology/Real Number Line",
"Cartesian Product of Open Ordinal Space with Uncountable Cartesia... |
proofwiki-13875 | Countably Compact Space is not necessarily Sequentially Compact | A countably compact topological space is not necessarily also a sequentially compact space. | Let $T := \hointr 0 \Omega \times \Bbb I^\Bbb I$ be the Cartesian product of $\hointr 0 \Omega$ under the interval topology, with the uncountable Cartesian product of the closed unit interval under the usual (Euclidean) topology.
From Cartesian Product of Open Ordinal Space with Uncountable Cartesian Product of Closed ... | A [[Definition:Countably Compact Space|countably compact topological space]] is not necessarily also a [[Definition:Sequentially Compact Space|sequentially compact space]]. | Let $T := \hointr 0 \Omega \times \Bbb I^\Bbb I$ be the [[Definition:Cartesian Product|Cartesian product]] of $\hointr 0 \Omega$ under the [[Definition:Interval Topology|interval topology]], with the [[Definition:Uncountable Set|uncountable]] [[Definition:Cartesian Product|Cartesian product]] of the [[Definition:Closed... | Countably Compact Space is not necessarily Sequentially Compact/Proof 1 | https://proofwiki.org/wiki/Countably_Compact_Space_is_not_necessarily_Sequentially_Compact | https://proofwiki.org/wiki/Countably_Compact_Space_is_not_necessarily_Sequentially_Compact/Proof_1 | [
"Countably Compact Space is not necessarily Sequentially Compact",
"Sequentially Compact Spaces",
"Countably Compact Spaces",
"Sequence of Implications of Global Compactness Properties"
] | [
"Definition:Countably Compact Space",
"Definition:Sequentially Compact Space"
] | [
"Definition:Cartesian Product",
"Definition:Order Topology",
"Definition:Uncountable/Set",
"Definition:Cartesian Product",
"Definition:Real Interval/Unit Interval/Closed",
"Definition:Euclidean Space/Euclidean Topology/Real Number Line",
"Cartesian Product of Open Ordinal Space with Uncountable Cartesia... |
proofwiki-13876 | Weakly Countably Compact Space is not necessarily Countably Compact | A weakly countably compact topological space is not also necessarily a countably compact space. | Let $T$ be the deleted integer topological space.
From Deleted Integer Topology is Weakly Countably Compact, $T$ is a weakly countably compact space.
From Deleted Integer Topology is not Countably Compact, $T$ is not a countably compact space.
Hence the result.
{{qed}} | A [[Definition:Weakly Countably Compact Space|weakly countably compact topological space]] is not also necessarily a [[Definition:Countably Compact Space|countably compact space]]. | Let $T$ be the [[Definition:Deleted Integer Topology|deleted integer topological space]].
From [[Deleted Integer Topology is Weakly Countably Compact]], $T$ is a [[Definition:Weakly Countably Compact Space|weakly countably compact space]].
From [[Deleted Integer Topology is not Countably Compact]], $T$ is not a [[De... | Weakly Countably Compact Space is not necessarily Countably Compact/Proof 1 | https://proofwiki.org/wiki/Weakly_Countably_Compact_Space_is_not_necessarily_Countably_Compact | https://proofwiki.org/wiki/Weakly_Countably_Compact_Space_is_not_necessarily_Countably_Compact/Proof_1 | [
"Weakly Countably Compact Space is not necessarily Countably Compact",
"Weakly Countably Compact Spaces",
"Countably Compact Spaces",
"Sequence of Implications of Global Compactness Properties"
] | [
"Definition:Weakly Countably Compact Space",
"Definition:Countably Compact Space"
] | [
"Definition:Deleted Integer Topology",
"Deleted Integer Topology is Weakly Countably Compact",
"Definition:Weakly Countably Compact Space",
"Deleted Integer Topology is not Countably Compact",
"Definition:Countably Compact Space"
] |
proofwiki-13877 | Weakly Countably Compact Space is not necessarily Countably Compact | A weakly countably compact topological space is not also necessarily a countably compact space. | Let $T$ be the odd-even topology on the strictly positive integers $\Z_{>0}$.
From Odd-Even Topology is Weakly Countably Compact, $T$ is a weakly countably compact space.
From Odd-Even Topology is not Countably Compact, $T$ is not a countably compact space.
Hence the result.
{{qed}} | A [[Definition:Weakly Countably Compact Space|weakly countably compact topological space]] is not also necessarily a [[Definition:Countably Compact Space|countably compact space]]. | Let $T$ be the [[Definition:Odd-Even Topology|odd-even topology]] on the [[Definition:Strictly Positive Integer|strictly positive integers]] $\Z_{>0}$.
From [[Odd-Even Topology is Weakly Countably Compact]], $T$ is a [[Definition:Weakly Countably Compact Space|weakly countably compact space]].
From [[Odd-Even Topolo... | Weakly Countably Compact Space is not necessarily Countably Compact/Proof 2 | https://proofwiki.org/wiki/Weakly_Countably_Compact_Space_is_not_necessarily_Countably_Compact | https://proofwiki.org/wiki/Weakly_Countably_Compact_Space_is_not_necessarily_Countably_Compact/Proof_2 | [
"Weakly Countably Compact Space is not necessarily Countably Compact",
"Weakly Countably Compact Spaces",
"Countably Compact Spaces",
"Sequence of Implications of Global Compactness Properties"
] | [
"Definition:Weakly Countably Compact Space",
"Definition:Countably Compact Space"
] | [
"Definition:Odd-Even Topology",
"Definition:Strictly Positive/Integer",
"Odd-Even Topology is Weakly Countably Compact",
"Definition:Weakly Countably Compact Space",
"Odd-Even Topology is not Countably Compact",
"Definition:Countably Compact Space"
] |
proofwiki-13878 | Pseudocompact Space is not necessarily Countably Compact | A pseudocompact topological space is not necessarily also a countably compact space. | Let $T$ be an infinite particular point space.
From Particular Point Space is Pseudocompact, $T$ is a pseudocompact space.
From Infinite Particular Point Space is not Weakly Countably Compact, $T$ is not a weakly countably compact space.
From Countably Compact Space is Weakly Countably Compact, $T$ cannot be a countabl... | A [[Definition:Pseudocompact Space|pseudocompact topological space]] is not necessarily also a [[Definition:Countably Compact Space|countably compact space]]. | Let $T$ be an [[Definition:Infinite Particular Point Topology|infinite particular point space]].
From [[Particular Point Space is Pseudocompact]], $T$ is a [[Definition:Pseudocompact Space|pseudocompact space]].
From [[Infinite Particular Point Space is not Weakly Countably Compact]], $T$ is not a [[Definition:Weakl... | Pseudocompact Space is not necessarily Countably Compact | https://proofwiki.org/wiki/Pseudocompact_Space_is_not_necessarily_Countably_Compact | https://proofwiki.org/wiki/Pseudocompact_Space_is_not_necessarily_Countably_Compact | [
"Pseudocompact Spaces",
"Countably Compact Spaces",
"Sequence of Implications of Global Compactness Properties"
] | [
"Definition:Pseudocompact Space",
"Definition:Countably Compact Space"
] | [
"Definition:Particular Point Topology/Infinite",
"Particular Point Space is Pseudocompact",
"Definition:Pseudocompact Space",
"Infinite Particular Point Space is not Weakly Countably Compact",
"Definition:Weakly Countably Compact Space",
"Countably Compact Space is Weakly Countably Compact",
"Definition... |
proofwiki-13879 | Weakly Locally Compact Space is not necessarily Strongly Locally Compact | A weakly locally compact topological space is not necessarily also a strongly locally compact space. | Let $T$ be the nested interval topological space.
From Nested Interval Space is Weakly Locally Compact, $T$ is a weakly locally compact space.
From Nested Interval Space is not Strongly Locally Compact, $T$ is not a strongly locally compact space.
Hence the result.
{{qed}} | A [[Definition:Weakly Locally Compact Space|weakly locally compact topological space]] is not necessarily also a [[Definition:Strongly Locally Compact Space|strongly locally compact space]]. | Let $T$ be the [[Definition:Nested Interval Space|nested interval topological space]].
From [[Nested Interval Space is Weakly Locally Compact]], $T$ is a [[Definition:Weakly Locally Compact Space|weakly locally compact space]].
From [[Nested Interval Space is not Strongly Locally Compact]], $T$ is not a [[Definition... | Weakly Locally Compact Space is not necessarily Strongly Locally Compact/Proof 1 | https://proofwiki.org/wiki/Weakly_Locally_Compact_Space_is_not_necessarily_Strongly_Locally_Compact | https://proofwiki.org/wiki/Weakly_Locally_Compact_Space_is_not_necessarily_Strongly_Locally_Compact/Proof_1 | [
"Weakly Locally Compact Space is not necessarily Strongly Locally Compact",
"Weakly Locally Compact Spaces",
"Strongly Locally Compact Spaces",
"Sequence of Implications of Local Compactness Properties"
] | [
"Definition:Weakly Locally Compact Space",
"Definition:Strongly Locally Compact Space"
] | [
"Definition:Nested Interval Topology",
"Nested Interval Space is Weakly Locally Compact",
"Definition:Weakly Locally Compact Space",
"Nested Interval Space is not Strongly Locally Compact",
"Definition:Strongly Locally Compact Space"
] |
proofwiki-13880 | Weakly Locally Compact Space is not necessarily Strongly Locally Compact | A weakly locally compact topological space is not necessarily also a strongly locally compact space. | Let $T$ be an infinite particular point space.
From Particular Point Space is Weakly Locally Compact, $T$ is a weakly locally compact space.
From Infinite Particular Point Space is not Strongly Locally Compact, $T$ is not a strongly locally compact space.
Hence the result.
{{qed}} | A [[Definition:Weakly Locally Compact Space|weakly locally compact topological space]] is not necessarily also a [[Definition:Strongly Locally Compact Space|strongly locally compact space]]. | Let $T$ be an [[Definition:Infinite Particular Point Topology|infinite particular point space]].
From [[Particular Point Space is Weakly Locally Compact]], $T$ is a [[Definition:Weakly Locally Compact Space|weakly locally compact space]].
From [[Infinite Particular Point Space is not Strongly Locally Compact]], $T$ ... | Weakly Locally Compact Space is not necessarily Strongly Locally Compact/Proof 2 | https://proofwiki.org/wiki/Weakly_Locally_Compact_Space_is_not_necessarily_Strongly_Locally_Compact | https://proofwiki.org/wiki/Weakly_Locally_Compact_Space_is_not_necessarily_Strongly_Locally_Compact/Proof_2 | [
"Weakly Locally Compact Space is not necessarily Strongly Locally Compact",
"Weakly Locally Compact Spaces",
"Strongly Locally Compact Spaces",
"Sequence of Implications of Local Compactness Properties"
] | [
"Definition:Weakly Locally Compact Space",
"Definition:Strongly Locally Compact Space"
] | [
"Definition:Particular Point Topology/Infinite",
"Particular Point Space is Weakly Locally Compact",
"Definition:Weakly Locally Compact Space",
"Infinite Particular Point Space is not Strongly Locally Compact",
"Definition:Strongly Locally Compact Space"
] |
proofwiki-13881 | Strongly Locally Compact Space is not necessarily Sigma-Compact | A strongly locally compact topological space is not necessarily also a $\sigma$-compact space. | Let $T$ be an uncountable discrete space.
From Discrete Space is Strongly Locally Compact, $T$ is a strongly locally compact space.
From Uncountable Discrete Space is not $\sigma$-Compact, $T$ is not a $\sigma$-compact space.
Hence the result.
{{qed}}
Category:Strongly Locally Compact Spaces
Category:Sigma-Compact Spac... | A [[Definition:Strongly Locally Compact Space|strongly locally compact topological space]] is not necessarily also a [[Definition:Sigma-Compact Space|$\sigma$-compact space]]. | Let $T$ be an [[Definition:Uncountable Discrete Space|uncountable discrete space]].
From [[Discrete Space is Strongly Locally Compact]], $T$ is a [[Definition:Strongly Locally Compact Space|strongly locally compact space]].
From [[Uncountable Discrete Space is not Sigma-Compact|Uncountable Discrete Space is not $\si... | Strongly Locally Compact Space is not necessarily Sigma-Compact | https://proofwiki.org/wiki/Strongly_Locally_Compact_Space_is_not_necessarily_Sigma-Compact | https://proofwiki.org/wiki/Strongly_Locally_Compact_Space_is_not_necessarily_Sigma-Compact | [
"Strongly Locally Compact Spaces",
"Sigma-Compact Spaces",
"Sequence of Implications of Local Compactness Properties"
] | [
"Definition:Strongly Locally Compact Space",
"Definition:Sigma-Compact Space"
] | [
"Definition:Discrete Topology/Uncountable",
"Discrete Space is Strongly Locally Compact",
"Definition:Strongly Locally Compact Space",
"Uncountable Discrete Space is not Sigma-Compact",
"Definition:Sigma-Compact Space",
"Category:Strongly Locally Compact Spaces",
"Category:Sigma-Compact Spaces",
"Cate... |
proofwiki-13882 | Weakly Sigma-Locally Compact Space is not necessarily Strongly Locally Compact | A weakly $\sigma$-locally compact topological space is not necessarily also a strongly locally compact space. | Let $T$ be the nested interval topological space.
From Nested Interval Space is Weakly $\sigma$-Locally Compact, $T$ is a weakly $\sigma$-locally compact space.
From Nested Interval Space is not Strongly Locally Compact, $T$ is not a strongly locally compact space.
Hence the result.
{{qed}} | A [[Definition:Weakly Sigma-Locally Compact Space|weakly $\sigma$-locally compact topological space]] is not necessarily also a [[Definition:Strongly Locally Compact Space|strongly locally compact space]]. | Let $T$ be the [[Definition:Nested Interval Space|nested interval topological space]].
From [[Nested Interval Space is Weakly Sigma-Locally Compact|Nested Interval Space is Weakly $\sigma$-Locally Compact]], $T$ is a [[Definition:Weakly Sigma-Locally Compact Space|weakly $\sigma$-locally compact space]].
From [[Nest... | Weakly Sigma-Locally Compact Space is not necessarily Strongly Locally Compact | https://proofwiki.org/wiki/Weakly_Sigma-Locally_Compact_Space_is_not_necessarily_Strongly_Locally_Compact | https://proofwiki.org/wiki/Weakly_Sigma-Locally_Compact_Space_is_not_necessarily_Strongly_Locally_Compact | [
"Strongly Locally Compact Spaces",
"Weakly Sigma-Locally Compact Spaces",
"Sequence of Implications of Local Compactness Properties"
] | [
"Definition:Weakly Sigma-Locally Compact Space",
"Definition:Strongly Locally Compact Space"
] | [
"Definition:Nested Interval Topology",
"Nested Interval Space is Weakly Sigma-Locally Compact",
"Definition:Weakly Sigma-Locally Compact Space",
"Nested Interval Space is not Strongly Locally Compact",
"Definition:Strongly Locally Compact Space"
] |
proofwiki-13883 | First-Countable Space is not necessarily Second-Countable | A first-countable topological space is not necessarily also a second-countable space. | Let $T$ be the Sorgenfrey line.
From Sorgenfrey Line is First-Countable, $T$ is a first-countable space.
From Sorgenfrey Line is not Second-Countable, $T$ is not a second-countable space.
Hence the result.
{{qed}} | A [[Definition:First-Countable Space|first-countable topological space]] is not necessarily also a [[Definition:Second-Countable Space|second-countable space]]. | Let $T$ be the [[Definition:Sorgenfrey Line|Sorgenfrey line]].
From [[Sorgenfrey Line is First-Countable]], $T$ is a [[Definition:First-Countable Space|first-countable space]].
From [[Sorgenfrey Line is not Second-Countable]], $T$ is not a [[Definition:Second-Countable Space|second-countable space]].
Hence the resu... | First-Countable Space is not necessarily Second-Countable | https://proofwiki.org/wiki/First-Countable_Space_is_not_necessarily_Second-Countable | https://proofwiki.org/wiki/First-Countable_Space_is_not_necessarily_Second-Countable | [
"First-Countable Spaces",
"Second-Countable Spaces"
] | [
"Definition:First-Countable Space",
"Definition:Second-Countable Space"
] | [
"Definition:Sorgenfrey Line",
"Sorgenfrey Line is First-Countable",
"Definition:First-Countable Space",
"Sorgenfrey Line is not Second-Countable",
"Definition:Second-Countable Space"
] |
proofwiki-13884 | Lindelöf Space is not necessarily Second-Countable | A Lindelöf space is not necessarily also a second-countable space. | Let $T$ be the Sorgenfrey line.
From Sorgenfrey Line is Lindelöf, $T$ is a Lindelöf space.
From Sorgenfrey Line is not Second-Countable, $T$ is not a second-countable space.
Hence the result.
{{qed}} | A [[Definition:Lindelöf Space|Lindelöf space]] is not necessarily also a [[Definition:Second-Countable Space|second-countable space]]. | Let $T$ be the [[Definition:Sorgenfrey Line|Sorgenfrey line]].
From [[Sorgenfrey Line is Lindelöf]], $T$ is a [[Definition:Lindelöf Space|Lindelöf space]].
From [[Sorgenfrey Line is not Second-Countable]], $T$ is not a [[Definition:Second-Countable Space|second-countable space]].
Hence the result.
{{qed}} | Lindelöf Space is not necessarily Second-Countable | https://proofwiki.org/wiki/Lindelöf_Space_is_not_necessarily_Second-Countable | https://proofwiki.org/wiki/Lindelöf_Space_is_not_necessarily_Second-Countable | [
"Lindelöf Spaces",
"Second-Countable Spaces"
] | [
"Definition:Lindelöf Space",
"Definition:Second-Countable Space"
] | [
"Definition:Sorgenfrey Line",
"Sorgenfrey Line is Lindelöf",
"Definition:Lindelöf Space",
"Sorgenfrey Line is not Second-Countable",
"Definition:Second-Countable Space"
] |
proofwiki-13885 | Separable Space is not necessarily Second-Countable | A separable topological space is not necessarily also a second-countable space. | Let $T$ be the Sorgenfrey line.
From Sorgenfrey Line is Separable, $T$ is a separable space.
From Sorgenfrey Line is not Second-Countable, $T$ is not a second-countable space.
Hence the result.
{{qed}} | A [[Definition:Separable Space|separable topological space]] is not necessarily also a [[Definition:Second-Countable Space|second-countable space]]. | Let $T$ be the [[Definition:Sorgenfrey Line|Sorgenfrey line]].
From [[Sorgenfrey Line is Separable]], $T$ is a [[Definition:Separable Space|separable space]].
From [[Sorgenfrey Line is not Second-Countable]], $T$ is not a [[Definition:Second-Countable Space|second-countable space]].
Hence the result.
{{qed}} | Separable Space is not necessarily Second-Countable | https://proofwiki.org/wiki/Separable_Space_is_not_necessarily_Second-Countable | https://proofwiki.org/wiki/Separable_Space_is_not_necessarily_Second-Countable | [
"Separable Spaces",
"Second-Countable Spaces"
] | [
"Definition:Separable Space",
"Definition:Second-Countable Space"
] | [
"Definition:Sorgenfrey Line",
"Sorgenfrey Line is Separable",
"Definition:Separable Space",
"Sorgenfrey Line is not Second-Countable",
"Definition:Second-Countable Space"
] |
proofwiki-13886 | Topological Space satisfying Countable Chain Condition is not necessarily Separable | A topological space which satisfies the countable chain condition is not necessarily also a separable space. | Let $T$ be a countable complement topology on an uncountable set.
From Countable Complement Space Satisfies Countable Chain Condition, $T$ satisfies the countable chain condition.
From Countable Complement Space is not Separable, $T$ is not a separable space.
Hence the result.
{{qed}} | A [[Definition:Topological Space|topological space]] which satisfies the [[Definition:Countable Chain Condition|countable chain condition]] is not necessarily also a [[Definition:Separable Space|separable space]]. | Let $T$ be a [[Definition:Countable Complement Topology|countable complement topology]] on an [[Definition:Uncountable Set|uncountable set]].
From [[Countable Complement Space Satisfies Countable Chain Condition]], $T$ satisfies the [[Definition:Countable Chain Condition|countable chain condition]].
From [[Countable... | Topological Space satisfying Countable Chain Condition is not necessarily Separable | https://proofwiki.org/wiki/Topological_Space_satisfying_Countable_Chain_Condition_is_not_necessarily_Separable | https://proofwiki.org/wiki/Topological_Space_satisfying_Countable_Chain_Condition_is_not_necessarily_Separable | [
"Countable Chain Condition",
"Separable Spaces"
] | [
"Definition:Topological Space",
"Definition:Countable Chain Condition",
"Definition:Separable Space"
] | [
"Definition:Countable Complement Topology",
"Definition:Uncountable/Set",
"Countable Complement Space Satisfies Countable Chain Condition",
"Definition:Countable Chain Condition",
"Countable Complement Space is not Separable",
"Definition:Separable Space"
] |
proofwiki-13887 | Equivalence of Definitions of Strongly Locally Compact Space | Let $T = \struct {S, \tau}$ be a topological space.
{{TFAE|def = Strongly Locally Compact Space}} | === 1 implies 2 ===
Follows immediately from Topological Closure is Closed.
{{qed|lemma}} | Let $T = \struct {S, \tau}$ be a [[Definition:Topological Space|topological space]].
{{TFAE|def = Strongly Locally Compact Space}} | === 1 implies 2 ===
Follows immediately from [[Topological Closure is Closed]].
{{qed|lemma}} | Equivalence of Definitions of Strongly Locally Compact Space | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Strongly_Locally_Compact_Space | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Strongly_Locally_Compact_Space | [
"Strongly Locally Compact Spaces"
] | [
"Definition:Topological Space"
] | [
"Topological Closure is Closed",
"Topological Closure is Closed"
] |
proofwiki-13888 | Uniqueness of Representing Objects | Let $C$ be a locally small category.
Let $\mathbf{Set}$ be the category of sets.
Let $F : \mathbf C \to \mathbf{Set}$ be a covariant functor.
Let $(A, \eta)$ and $(B, \xi)$ be representations of $F$.
Then there exists a unique isomorphism $f : A \to B$ such that $\eta \circ h^f = \xi$, where:
:$h^f$ is the precompositi... | As $\eta : \map {\operatorname {Hom} } {A, \cdot} \to F$ is a natural isomorphism, it has an inverse:
:$\eta' : F \to \map {\operatorname {Hom} } {A, \cdot}$
Similarly, $\xi : \map {\operatorname {Hom} } {B, \cdot} \to F$ also has an inverse:
:$\xi' : F \to \map {\operatorname {Hom} } {B, \cdot}$
Let $\phi = \eta' \cir... | Let $C$ be a [[Definition:Locally Small Category|locally small category]].
Let $\mathbf{Set}$ be the [[Definition:Category of Sets|category of sets]].
Let $F : \mathbf C \to \mathbf{Set}$ be a [[Definition:Covariant Functor|covariant functor]].
Let $(A, \eta)$ and $(B, \xi)$ be [[Definition:Representation of Functor... | As $\eta : \map {\operatorname {Hom} } {A, \cdot} \to F$ is a [[Definition:Natural Isomorphism|natural isomorphism]], it has an [[Definition:Inverse Natural Isomorphism between Covariant Functors|inverse]]:
:$\eta' : F \to \map {\operatorname {Hom} } {A, \cdot}$
Similarly, $\xi : \map {\operatorname {Hom} } {B, \cdot}... | Uniqueness of Representing Objects | https://proofwiki.org/wiki/Uniqueness_of_Representing_Objects | https://proofwiki.org/wiki/Uniqueness_of_Representing_Objects | [
"Functors"
] | [
"Definition:Locally Small Category",
"Definition:Category of Sets",
"Definition:Functor/Covariant",
"Definition:Representation of Functor",
"Definition:Unique",
"Definition:Isomorphism (Category Theory)",
"Definition:Precomposition Natural Transformation",
"Definition:Vertical Composition of Natural T... | [
"Definition:Natural Isomorphism",
"Definition:Natural Isomorphism between Covariant Functors/Inverse",
"Definition:Natural Isomorphism between Covariant Functors/Inverse",
"Definition:Vertical Composition of Natural Transformations",
"Vertical Composition of Natural Transformations is Natural Transformation... |
proofwiki-13889 | Existence and Uniqueness of Reduced Form of Group Word | Let $X$ be a set.
Let $w$ be a group word on $X$.
Then $w$ has a unique reduced form. | === Existence ===
By induction on the length of $w$. | Let $X$ be a [[Definition:Set|set]].
Let $w$ be a [[Definition:Group Word on Set|group word]] on $X$.
Then $w$ has a [[Definition:Unique|unique]] [[Definition:Reduced Form of Group Word|reduced form]]. | === Existence ===
By induction on the [[Definition:Length of Ordered Tuple|length]] of $w$. | Existence and Uniqueness of Reduced Form of Group Word | https://proofwiki.org/wiki/Existence_and_Uniqueness_of_Reduced_Form_of_Group_Word | https://proofwiki.org/wiki/Existence_and_Uniqueness_of_Reduced_Form_of_Group_Word | [
"Group Words",
"Proofs by Induction"
] | [
"Definition:Set",
"Definition:Group Word on Set",
"Definition:Unique",
"Definition:Reduced Form of Group Word"
] | [
"Definition:Length of Sequence",
"Definition:Length of Sequence"
] |
proofwiki-13890 | Abelianization of Free Group is Free Abelian Group | Let $X$ be a set.
Let $\struct {F_X, \iota}$ be a free group on $X$.
Let $F_X^{\mathrm {ab} }$ be its abelianization.
Let $\pi : F_X \to F_X^{\mathrm {ab} }$ be the quotient group epimorphism.
Then $\struct {F_X^{\mathrm {ab} }, \pi \circ \iota}$ is a free abelian group on $X$. | :$\xymatrix {
X \ar[d]_\iota \ar[rd]^{\forall f} & \\
F_X \ar[d]_\pi \ar[r]^{\exists ! g} & H \\
F_X^{\mathrm {ab} } \ar[ru]_{\exists ! h} }$
Lef $H$ be any abelian group and $f:X \to H$ be any mapping.
By Universal Property of Free Group on Set, there exists a unique group homomorphism $g:F_X \to H$ such that $g \circ... | Let $X$ be a [[Definition:Set|set]].
Let $\struct {F_X, \iota}$ be a [[Definition:Free Group on Set|free group]] on $X$.
Let $F_X^{\mathrm {ab} }$ be its [[Definition:Abelianization of Group|abelianization]].
Let $\pi : F_X \to F_X^{\mathrm {ab} }$ be the [[Definition:Quotient Group Epimorphism|quotient group epimor... | :$\xymatrix {
X \ar[d]_\iota \ar[rd]^{\forall f} & \\
F_X \ar[d]_\pi \ar[r]^{\exists ! g} & H \\
F_X^{\mathrm {ab} } \ar[ru]_{\exists ! h} }$
Lef $H$ be any [[Definition:Abelian Group|abelian group]] and $f:X \to H$ be any mapping.
By [[Definition:Universal Property of Free Group on Set|Universal Property of Free Gro... | Abelianization of Free Group is Free Abelian Group | https://proofwiki.org/wiki/Abelianization_of_Free_Group_is_Free_Abelian_Group | https://proofwiki.org/wiki/Abelianization_of_Free_Group_is_Free_Abelian_Group | [
"Group Theory"
] | [
"Definition:Set",
"Definition:Free Group on Set",
"Definition:Abelianization of Group",
"Definition:Quotient Epimorphism/Group",
"Definition:Free Abelian Group on Set"
] | [
"Definition:Abelian Group",
"Definition:Free Group on Set",
"Definition:Unique",
"Definition:Group Homomorphism",
"Universal Property of Abelianization of Group",
"Definition:Unique",
"Definition:Group Homomorphism",
"Universal Property of Free Abelian Group on Set",
"Category:Group Theory"
] |
proofwiki-13891 | Paracompact Space is not necessarily Compact | A paracompact topological space is not necessarily also a compact topological space. | Let $T = \struct {\R, \tau_d}$ be the real number line with the usual (Euclidean) topology.
From Real Number Line is Paracompact, $T$ is a paracompact space.
From Real Number Line is not Compact, $T$ is not a compact space.
Hence the result.
{{qed}} | A [[Definition:Paracompact Space|paracompact topological space]] is not necessarily also a [[Definition:Compact Topological Space|compact topological space]]. | Let $T = \struct {\R, \tau_d}$ be the [[Definition:Real Number Line with Euclidean Topology|real number line with the usual (Euclidean) topology]].
From [[Real Number Line is Paracompact]], $T$ is a [[Definition:Paracompact Space|paracompact space]].
From [[Real Number Line is not Compact]], $T$ is not a [[Definitio... | Paracompact Space is not necessarily Compact | https://proofwiki.org/wiki/Paracompact_Space_is_not_necessarily_Compact | https://proofwiki.org/wiki/Paracompact_Space_is_not_necessarily_Compact | [
"Paracompact Spaces",
"Compact Topological Spaces",
"Sequence of Implications of Paracompactness Properties"
] | [
"Definition:Paracompact Space",
"Definition:Compact Topological Space"
] | [
"Definition:Euclidean Space/Euclidean Topology/Real Number Line",
"Real Number Line is Paracompact",
"Definition:Paracompact Space",
"Real Number Line is not Compact",
"Definition:Compact Topological Space"
] |
proofwiki-13892 | Metacompact Space is not necessarily Paracompact | A metacompact topological space is not necessarily also a paracompact space. | Let $T$ be the Dieudonné plank.
From Dieudonné Plank is Metacompact, $T$ is a metacompact space.
From Dieudonné Plank is not Paracompact, $T$ is not a paracompact space.
Hence the result.
{{qed}} | A [[Definition:Metacompact Space|metacompact topological space]] is not necessarily also a [[Definition:Paracompact Space|paracompact space]]. | Let $T$ be the [[Definition:Dieudonné Plank|Dieudonné plank]].
From [[Dieudonné Plank is Metacompact]], $T$ is a [[Definition:Metacompact Space|metacompact space]].
From [[Dieudonné Plank is not Paracompact]], $T$ is not a [[Definition:Paracompact Space|paracompact space]].
Hence the result.
{{qed}} | Metacompact Space is not necessarily Paracompact | https://proofwiki.org/wiki/Metacompact_Space_is_not_necessarily_Paracompact | https://proofwiki.org/wiki/Metacompact_Space_is_not_necessarily_Paracompact | [
"Paracompact Spaces",
"Metacompact Spaces",
"Sequence of Implications of Paracompactness Properties"
] | [
"Definition:Metacompact Space",
"Definition:Paracompact Space"
] | [
"Definition:Dieudonné Plank",
"Dieudonné Plank is Metacompact",
"Definition:Metacompact Space",
"Dieudonné Plank is not Paracompact",
"Definition:Paracompact Space"
] |
proofwiki-13893 | Paracompact Countably Compact Space is Compact | Let $T = \struct {S, \tau}$ be a countably compact space which is also paracompact.
Then $T$ is compact. | From the definition of paracompact space, a paracompact space is also a metacompact space.
The result follows from Metacompact Countably Compact Space is Compact.
{{qed}} | Let $T = \struct {S, \tau}$ be a [[Definition:Countably Compact Space|countably compact space]] which is also [[Definition:Paracompact Space|paracompact]].
Then $T$ is [[Definition:Compact Topological Space|compact]]. | From the definition of [[Definition:Paracompact Space|paracompact space]], a [[Definition:Paracompact Space|paracompact space]] is also a [[Definition:Metacompact Space|metacompact space]].
The result follows from [[Metacompact Countably Compact Space is Compact]].
{{qed}} | Paracompact Countably Compact Space is Compact | https://proofwiki.org/wiki/Paracompact_Countably_Compact_Space_is_Compact | https://proofwiki.org/wiki/Paracompact_Countably_Compact_Space_is_Compact | [
"Paracompact Spaces",
"Countably Compact Spaces",
"Compact Topological Spaces"
] | [
"Definition:Countably Compact Space",
"Definition:Paracompact Space",
"Definition:Compact Topological Space"
] | [
"Definition:Paracompact Space",
"Definition:Paracompact Space",
"Definition:Metacompact Space",
"Metacompact Countably Compact Space is Compact"
] |
proofwiki-13894 | Minimal Hausdorff Space is not necessarily Compact | Let $S$ be a set.
Let $\tau$ be the minimal subset of the power set $\powerset S$ such that $\struct {S, \tau}$ is a Hausdorff space.
Then it is not necessarily the case that $\struct {S, \tau}$ is compact. | Let $T = \struct {S, \tau}$ be the canonical minimal Hausdorff non-compact space.
This space has been so named on {{ProofWiki}} in order to allow reference to it without needing to describe it whenever it is mentioned.
By Canonical Minimal Hausdorff Non-Compact Space is Minimal Hausdorff, $\tau$ is the minimal subset o... | Let $S$ be a [[Definition:Set|set]].
Let $\tau$ be the [[Definition:Minimal Set|minimal]] [[Definition:Subset|subset]] of the [[Definition:Power Set|power set]] $\powerset S$ such that $\struct {S, \tau}$ is a [[Definition:Hausdorff Space|Hausdorff space]].
Then it is not necessarily the case that $\struct {S, \tau}... | Let $T = \struct {S, \tau}$ be the [[Definition:Canonical Minimal Hausdorff Non-Compact Space|canonical minimal Hausdorff non-compact space]].
This space has been so named on {{ProofWiki}} in order to allow reference to it without needing to describe it whenever it is mentioned.
By [[Canonical Minimal Hausdorff Non-C... | Minimal Hausdorff Space is not necessarily Compact | https://proofwiki.org/wiki/Minimal_Hausdorff_Space_is_not_necessarily_Compact | https://proofwiki.org/wiki/Minimal_Hausdorff_Space_is_not_necessarily_Compact | [
"Hausdorff Spaces",
"Compact Topological Spaces"
] | [
"Definition:Set",
"Definition:Minimal/Set",
"Definition:Subset",
"Definition:Power Set",
"Definition:T2 Space",
"Definition:Compact Topological Space"
] | [
"Definition:Canonical Minimal Hausdorff Non-Compact Space",
"Canonical Minimal Hausdorff Non-Compact Space is Minimal Hausdorff",
"Definition:Minimal/Set",
"Definition:Subset",
"Definition:Power Set",
"Definition:T2 Space",
"Canonical Minimal Hausdorff Non-Compact Space is not Compact",
"Definition:Co... |
proofwiki-13895 | Maximal Compact Topological Space is not necessarily Hausdorff | Let $S$ be a set.
Let $\tau$ be the maximal subset of the power set $\powerset S$ such that $\struct {S, \tau}$ is a compact topological space.
Then it is not necessarily the case that $\struct {S, \tau}$ is a Hausdorff space. | Let $T = \struct {S, \tau}$ be the canonical maximal compact non-Hausdorff space.
This space has been so named on {{ProofWiki}} in order to allow reference to it without needing to describe it whenever it is mentioned.
By Canonical Maximal Compact Non-Hausdorff Space is Maximal Compact, $\tau$ is the maximal subset of ... | Let $S$ be a [[Definition:Set|set]].
Let $\tau$ be the [[Definition:Maximal Set|maximal]] [[Definition:Subset|subset]] of the [[Definition:Power Set|power set]] $\powerset S$ such that $\struct {S, \tau}$ is a [[Definition:Compact Topological Space|compact topological space]].
Then it is not necessarily the case that... | Let $T = \struct {S, \tau}$ be the [[Definition:Canonical Maximal Compact Non-Hausdorff Space|canonical maximal compact non-Hausdorff space]].
This space has been so named on {{ProofWiki}} in order to allow reference to it without needing to describe it whenever it is mentioned.
By [[Canonical Maximal Compact Non-Hau... | Maximal Compact Topological Space is not necessarily Hausdorff | https://proofwiki.org/wiki/Maximal_Compact_Topological_Space_is_not_necessarily_Hausdorff | https://proofwiki.org/wiki/Maximal_Compact_Topological_Space_is_not_necessarily_Hausdorff | [
"Hausdorff Spaces",
"Compact Topological Spaces"
] | [
"Definition:Set",
"Definition:Maximal/Set",
"Definition:Subset",
"Definition:Power Set",
"Definition:Compact Topological Space",
"Definition:T2 Space"
] | [
"Definition:Canonical Maximal Compact Non-Hausdorff Space",
"Canonical Maximal Compact Non-Hausdorff Space is Maximal Compact",
"Definition:Maximal/Set",
"Definition:Subset",
"Definition:Power Set",
"Definition:Compact Topological Space",
"Canonical Maximal Compact Non-Hausdorff Space is not Hausdorff",... |
proofwiki-13896 | Infinite Product of Sigma-Compact Spaces is not always Sigma-Compact | Let $I$ be an indexing set with infinite cardinality.
Let $\family {\struct {S_\alpha, \tau_\alpha} }_{\alpha \mathop \in I}$ be a family of topological spaces indexed by $I$.
Let $\ds \struct {S, \tau} = \prod_{\alpha \mathop \in I} \struct {S_\alpha, \tau_\alpha}$ be the product space of $\family {\struct {S_\alpha, ... | Let $T = \struct {\Z_{\ge 0}, \tau}$ be the topological space formed by the discrete topology on the set of positive integers.
Let $T' = \struct {\ds \prod_{\alpha \mathop \in \Z_{\ge 0} } \struct {\Z_{\ge 0}, \tau}_\alpha, \tau'}$ be the countable Cartesian product of $\struct {\Z_{\ge 0}, \tau}$ indexed by $\Z_{\ge 0... | Let $I$ be an [[Definition:Indexing Set|indexing set]] with [[Definition:Infinite Set|infinite cardinality]].
Let $\family {\struct {S_\alpha, \tau_\alpha} }_{\alpha \mathop \in I}$ be a [[Definition:Indexed Family|family]] of [[Definition:Topological Space|topological spaces]] [[Definition:Indexed Family|indexed]] by... | Let $T = \struct {\Z_{\ge 0}, \tau}$ be the [[Definition:Topological Space|topological space]] formed by the [[Definition:Discrete Topology|discrete topology]] on the [[Definition:Positive Integer|set of positive integers]].
Let $T' = \struct {\ds \prod_{\alpha \mathop \in \Z_{\ge 0} } \struct {\Z_{\ge 0}, \tau}_\alph... | Infinite Product of Sigma-Compact Spaces is not always Sigma-Compact | https://proofwiki.org/wiki/Infinite_Product_of_Sigma-Compact_Spaces_is_not_always_Sigma-Compact | https://proofwiki.org/wiki/Infinite_Product_of_Sigma-Compact_Spaces_is_not_always_Sigma-Compact | [
"Sigma-Compact Spaces",
"Product Topology"
] | [
"Definition:Indexing Set",
"Definition:Infinite Set",
"Definition:Indexing Set/Family",
"Definition:Topological Space",
"Definition:Indexing Set/Family",
"Definition:Product Space (Topology)",
"Definition:Sigma-Compact Space",
"Definition:Sigma-Compact Space"
] | [
"Definition:Topological Space",
"Definition:Discrete Topology",
"Definition:Positive/Integer",
"Definition:Cartesian Product/Countable",
"Definition:Indexing Set/Family",
"Definition:Product Topology",
"Countable Discrete Space is Sigma-Compact",
"Definition:Sigma-Compact Space",
"Countable Product ... |
proofwiki-13897 | Uncountable Product of Sequentially Compact Spaces is not always Sequentially Compact | Let $I$ be an indexing set with uncountable cardinality.
Let $\family {\struct {S_\alpha, \tau_\alpha} }_{\alpha \mathop \in I}$ be a family of topological spaces indexed by $I$.
Let $\ds \struct {S, \tau} = \prod_{\alpha \mathop \in I} \struct {S_\alpha, \tau_\alpha}$ be the product space of $\family {\struct {S_\alph... | Let $\mathbb I$ denote the closed unit interval.
Let $T = \struct {\mathbb I, \tau}$ be the topological space consisting of $\mathbb I$ under the usual (Euclidean) topology.
Let $T' = \mathbb I^{\mathbb I} = \struct {\ds \prod_{\alpha \mathop \in \mathbb I} \struct {\mathbb I, \tau}_\alpha, \tau'}$ be the uncountable C... | Let $I$ be an [[Definition:Indexing Set|indexing set]] with [[Definition:Uncountable Set|uncountable cardinality]].
Let $\family {\struct {S_\alpha, \tau_\alpha} }_{\alpha \mathop \in I}$ be a [[Definition:Indexed Family|family]] of [[Definition:Topological Space|topological spaces]] [[Definition:Indexed Family|indexe... | Let $\mathbb I$ denote the [[Definition:Closed Unit Interval|closed unit interval]].
Let $T = \struct {\mathbb I, \tau}$ be the [[Definition:Topological Space|topological space]] consisting of $\mathbb I$ under the [[Definition:Euclidean Topology on Real Number Line|usual (Euclidean) topology]].
Let $T' = \mathbb I^{... | Uncountable Product of Sequentially Compact Spaces is not always Sequentially Compact | https://proofwiki.org/wiki/Uncountable_Product_of_Sequentially_Compact_Spaces_is_not_always_Sequentially_Compact | https://proofwiki.org/wiki/Uncountable_Product_of_Sequentially_Compact_Spaces_is_not_always_Sequentially_Compact | [
"Sequentially Compact Spaces",
"Product Topology"
] | [
"Definition:Indexing Set",
"Definition:Uncountable/Set",
"Definition:Indexing Set/Family",
"Definition:Topological Space",
"Definition:Indexing Set/Family",
"Definition:Product Space (Topology)",
"Definition:Sequentially Compact Space",
"Definition:Sequentially Compact Space"
] | [
"Definition:Real Interval/Unit Interval/Closed",
"Definition:Topological Space",
"Definition:Euclidean Space/Euclidean Topology/Real Number Line",
"Definition:Cartesian Product/Uncountable",
"Definition:Indexing Set/Family",
"Definition:Product Topology",
"Closed Real Interval is Sequentially Compact",
... |
proofwiki-13898 | Product of Countably Compact Spaces is not always Countably Compact | Let $I$ be an indexing set.
Let $\family {\struct {S_\alpha, \tau_\alpha} }_{\alpha \mathop \in I}$ be a family of topological spaces indexed by $I$.
Let $\ds \struct {S, \tau} = \prod_{\alpha \mathop \in I} \struct {S_\alpha, \tau_\alpha}$ be the product space of $\family {\struct {S_\alpha, \tau_\alpha} }_{\alpha \ma... | Let $T$ denote the Novak space.
Let $T \times T$ denote the Cartesian product of the Novak space with itself under the product topology.
From Novak Space is Countably Compact, $T$ is a countably compact space.
But from Cartesian Product of Novak Spaces is not Countably Compact, $T \times T$ is not a countably compact s... | Let $I$ be an [[Definition:Indexing Set|indexing set]].
Let $\family {\struct {S_\alpha, \tau_\alpha} }_{\alpha \mathop \in I}$ be a [[Definition:Indexed Family|family]] of [[Definition:Topological Space|topological spaces]] [[Definition:Indexed Family|indexed]] by $I$.
Let $\ds \struct {S, \tau} = \prod_{\alpha \mat... | Let $T$ denote the [[Definition:Novak Space|Novak space]].
Let $T \times T$ denote the [[Definition:Cartesian Product|Cartesian product]] of the [[Definition:Novak Space|Novak space]] with itself under the [[Definition:Product Topology|product topology]].
From [[Novak Space is Countably Compact]], $T$ is a [[Definiti... | Product of Countably Compact Spaces is not always Countably Compact | https://proofwiki.org/wiki/Product_of_Countably_Compact_Spaces_is_not_always_Countably_Compact | https://proofwiki.org/wiki/Product_of_Countably_Compact_Spaces_is_not_always_Countably_Compact | [
"Countably Compact Spaces",
"Product Spaces"
] | [
"Definition:Indexing Set",
"Definition:Indexing Set/Family",
"Definition:Topological Space",
"Definition:Indexing Set/Family",
"Definition:Product Space (Topology)",
"Definition:Countably Compact Space",
"Definition:Countably Compact Space"
] | [
"Definition:Novak Space",
"Definition:Cartesian Product",
"Definition:Novak Space",
"Definition:Product Topology",
"Novak Space is Countably Compact",
"Definition:Countably Compact Space",
"Cartesian Product of Novak Spaces is not Countably Compact",
"Definition:Countably Compact Space"
] |
proofwiki-13899 | Infinite Product of Weakly Locally Compact Spaces is not always Weakly Locally Compact | Let $I$ be an indexing set with infinite cardinality.
Let $\family {\struct {S_\alpha, \tau_\alpha} }_{\alpha \mathop \in I}$ be a family of topological spaces indexed by $I$.
Let $\ds \struct {S, \tau} = \prod_{\alpha \mathop \in I} \struct {S_\alpha, \tau_\alpha}$ be the product space of $\family {\struct {S_\alpha, ... | Let $T = \struct {\Z_{\ge 0}, \tau}$ be the topological space formed by the discrete topology on the set of positive integers.
Let $T' = \struct {\ds \prod_{\alpha \mathop \in \Z_{\ge 0} } \struct {\Z_{\ge 0}, \tau}_\alpha, \tau'}$ be the countable Cartesian product of $\struct {\Z_{\ge 0}, \tau}$ indexed by $\Z_{\ge 0... | Let $I$ be an [[Definition:Indexing Set|indexing set]] with [[Definition:Infinite Set|infinite cardinality]].
Let $\family {\struct {S_\alpha, \tau_\alpha} }_{\alpha \mathop \in I}$ be a [[Definition:Indexed Family|family]] of [[Definition:Topological Space|topological spaces]] [[Definition:Indexed Family|indexed]] by... | Let $T = \struct {\Z_{\ge 0}, \tau}$ be the [[Definition:Topological Space|topological space]] formed by the [[Definition:Discrete Topology|discrete topology]] on the [[Definition:Positive Integer|set of positive integers]].
Let $T' = \struct {\ds \prod_{\alpha \mathop \in \Z_{\ge 0} } \struct {\Z_{\ge 0}, \tau}_\alph... | Infinite Product of Weakly Locally Compact Spaces is not always Weakly Locally Compact | https://proofwiki.org/wiki/Infinite_Product_of_Weakly_Locally_Compact_Spaces_is_not_always_Weakly_Locally_Compact | https://proofwiki.org/wiki/Infinite_Product_of_Weakly_Locally_Compact_Spaces_is_not_always_Weakly_Locally_Compact | [
"Weakly Locally Compact Spaces",
"Product Topology"
] | [
"Definition:Indexing Set",
"Definition:Infinite Set",
"Definition:Indexing Set/Family",
"Definition:Topological Space",
"Definition:Indexing Set/Family",
"Definition:Product Space (Topology)",
"Definition:Weakly Locally Compact Space",
"Definition:Weakly Locally Compact Space"
] | [
"Definition:Topological Space",
"Definition:Discrete Topology",
"Definition:Positive/Integer",
"Definition:Cartesian Product/Countable",
"Definition:Indexing Set/Family",
"Definition:Product Topology",
"Discrete Space is Strongly Locally Compact",
"Definition:Strongly Locally Compact Space",
"Strong... |
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