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-13900 | Product of Lindelöf Spaces is not always Lindelöf | 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 $T$ be the Sorgenfrey line.
Let $T' = T \times T$ be Sorgenfrey's half-open square topology.
From Sorgenfrey Line is Lindelöf, $T$ is a Lindelöf space.
From Sorgenfrey's Half-Open Square Topology is Not Lindelöf, $T'$ is not a Lindelöf space.
Hence the result.
{{qed}} | 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 $T$ be the [[Definition:Sorgenfrey Line|Sorgenfrey line]].
Let $T' = T \times T$ be [[Definition:Sorgenfrey's Half-Open Square Topology|Sorgenfrey's half-open square topology]].
From [[Sorgenfrey Line is Lindelöf]], $T$ is a [[Definition:Lindelöf Space|Lindelöf space]].
From [[Sorgenfrey's Half-Open Square Topol... | Product of Lindelöf Spaces is not always Lindelöf | https://proofwiki.org/wiki/Product_of_Lindelöf_Spaces_is_not_always_Lindelöf | https://proofwiki.org/wiki/Product_of_Lindelöf_Spaces_is_not_always_Lindelöf | [
"Lindelöf Spaces",
"Product Spaces"
] | [
"Definition:Indexing Set",
"Definition:Indexing Set/Family",
"Definition:Topological Space",
"Definition:Indexing Set/Family",
"Definition:Product Space (Topology)",
"Definition:Lindelöf Space",
"Definition:Lindelöf Space"
] | [
"Definition:Sorgenfrey Line",
"Definition:Sorgenfrey's Half-Open Square Topology",
"Sorgenfrey Line is Lindelöf",
"Definition:Lindelöf Space",
"Sorgenfrey's Half-Open Square Topology is Not Lindelöf",
"Definition:Lindelöf Space"
] |
proofwiki-13901 | Uncountable Product of First-Countable Spaces is not always First-Countable | 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 $T = \struct {\Z_{\ge 0}, \tau}$ denote the topological space consisting of the set of positive integers $\Z_{\ge 0}$ under the discrete topology.
Let $I$ be an indexing set with uncountable cardinality.
Let $T' = \struct {\ds \prod_{\alpha \mathop \in \mathbb I} \struct {\Z_{\ge 0}, \tau}_\alpha, \tau'}$ be the un... | 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 $T = \struct {\Z_{\ge 0}, \tau}$ denote the [[Definition:Topological Space|topological space]] consisting of the [[Definition:Positive Integer|set of positive integers]] $\Z_{\ge 0}$ under the [[Definition:Discrete Topology|discrete topology]].
Let $I$ be an [[Definition:Indexing Set|indexing set]] with [[Definiti... | Uncountable Product of First-Countable Spaces is not always First-Countable | https://proofwiki.org/wiki/Uncountable_Product_of_First-Countable_Spaces_is_not_always_First-Countable | https://proofwiki.org/wiki/Uncountable_Product_of_First-Countable_Spaces_is_not_always_First-Countable | [
"First-Countable Spaces",
"Product Topology",
"Uncountable Sets"
] | [
"Definition:Indexing Set",
"Definition:Uncountable/Set",
"Definition:Indexing Set/Family",
"Definition:Topological Space",
"Definition:Indexing Set/Family",
"Definition:Product Space (Topology)",
"Definition:First-Countable Space",
"Definition:First-Countable Space"
] | [
"Definition:Topological Space",
"Definition:Positive/Integer",
"Definition:Discrete Topology",
"Definition:Indexing Set",
"Definition:Uncountable/Set",
"Definition:Cartesian Product/Uncountable",
"Definition:Indexing Set/Family",
"Definition:Product Topology",
"Discrete Space is First-Countable",
... |
proofwiki-13902 | Uncountable Product of Second-Countable Spaces is not always Second-Countable | 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 $T = \struct {\Z_{\ge 0}, \tau}$ denote the topological space consisting of the set of positive integers $\Z_{\ge 0}$ under the discrete topology.
Let $I$ be an indexing set with uncountable cardinality.
Let $T' = \struct {\ds \prod_{\alpha \mathop \in I} \struct {\Z_{\ge 0}, \tau}_\alpha, \tau'}$ be the uncountabl... | 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 $T = \struct {\Z_{\ge 0}, \tau}$ denote the [[Definition:Topological Space|topological space]] consisting of the [[Definition:Positive Integer|set of positive integers]] $\Z_{\ge 0}$ under the [[Definition:Discrete Topology|discrete topology]].
Let $I$ be an [[Definition:Indexing Set|indexing set]] with [[Definiti... | Uncountable Product of Second-Countable Spaces is not always Second-Countable | https://proofwiki.org/wiki/Uncountable_Product_of_Second-Countable_Spaces_is_not_always_Second-Countable | https://proofwiki.org/wiki/Uncountable_Product_of_Second-Countable_Spaces_is_not_always_Second-Countable | [
"Second-Countable Spaces",
"Product Topology",
"Uncountable Sets"
] | [
"Definition:Indexing Set",
"Definition:Uncountable/Set",
"Definition:Indexing Set/Family",
"Definition:Topological Space",
"Definition:Indexing Set/Family",
"Definition:Product Space (Topology)",
"Definition:Second-Countable Space",
"Definition:Second-Countable Space"
] | [
"Definition:Topological Space",
"Definition:Positive/Integer",
"Definition:Discrete Topology",
"Definition:Indexing Set",
"Definition:Uncountable/Set",
"Definition:Cartesian Product/Uncountable",
"Definition:Indexing Set/Family",
"Definition:Product Topology",
"Countable Discrete Space is Second-Cou... |
proofwiki-13903 | Uncountable Product of Separable Spaces is not always Separable | 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 $T = \struct {\Z_{\ge 0}, \tau}$ denote the topological space consisting of the set of positive integers $\Z_{\ge 0}$ under the discrete topology.
Let $I$ be an indexing set with uncountable cardinality.
Let $\ds T' = \struct {\prod_{\alpha \mathop \in I} \struct {\Z_{\ge 0}, \tau}_\alpha, \tau'}$ be the uncountabl... | 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 $T = \struct {\Z_{\ge 0}, \tau}$ denote the [[Definition:Topological Space|topological space]] consisting of the [[Definition:Positive Integer|set of positive integers]] $\Z_{\ge 0}$ under the [[Definition:Discrete Topology|discrete topology]].
Let $I$ be an [[Definition:Indexing Set|indexing set]] with [[Definiti... | Uncountable Product of Separable Spaces is not always Separable | https://proofwiki.org/wiki/Uncountable_Product_of_Separable_Spaces_is_not_always_Separable | https://proofwiki.org/wiki/Uncountable_Product_of_Separable_Spaces_is_not_always_Separable | [
"Separable Spaces",
"Product Topology",
"Uncountable Sets"
] | [
"Definition:Indexing Set",
"Definition:Uncountable/Set",
"Definition:Indexing Set/Family",
"Definition:Topological Space",
"Definition:Indexing Set/Family",
"Definition:Product Space (Topology)",
"Definition:Separable Space",
"Definition:Separable Space"
] | [
"Definition:Topological Space",
"Definition:Positive/Integer",
"Definition:Discrete Topology",
"Definition:Indexing Set",
"Definition:Uncountable/Set",
"Definition:Cartesian Product/Uncountable",
"Definition:Indexing Set/Family",
"Definition:Product Topology",
"Countable Discrete Space is Separable"... |
proofwiki-13904 | Product of Paracompact Spaces is not always Paracompact | 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 $T$ be the Sorgenfrey line.
Let $T' = T \times T$ be Sorgenfrey's half-open square topology.
From Sorgenfrey Line is Paracompact, $T$ is a paracompact space.
From Sorgenfrey's Half-Open Square Topology is Not Paracompact, $T'$ is not a paracompact space.
Hence the result.
{{qed}} | 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 $T$ be the [[Definition:Sorgenfrey Line|Sorgenfrey line]].
Let $T' = T \times T$ be [[Definition:Sorgenfrey's Half-Open Square Topology|Sorgenfrey's half-open square topology]].
From [[Sorgenfrey Line is Paracompact]], $T$ is a [[Definition:Paracompact Space|paracompact space]].
From [[Sorgenfrey's Half-Open Squ... | Product of Paracompact Spaces is not always Paracompact | https://proofwiki.org/wiki/Product_of_Paracompact_Spaces_is_not_always_Paracompact | https://proofwiki.org/wiki/Product_of_Paracompact_Spaces_is_not_always_Paracompact | [
"Paracompact Spaces",
"Product Spaces"
] | [
"Definition:Indexing Set",
"Definition:Indexing Set/Family",
"Definition:Topological Space",
"Definition:Indexing Set/Family",
"Definition:Product Space (Topology)",
"Definition:Paracompact Space",
"Definition:Paracompact Space"
] | [
"Definition:Sorgenfrey Line",
"Definition:Sorgenfrey's Half-Open Square Topology",
"Sorgenfrey Line is Paracompact",
"Definition:Paracompact Space",
"Sorgenfrey's Half-Open Square Topology is Not Paracompact",
"Definition:Paracompact Space"
] |
proofwiki-13905 | Product of Metacompact Spaces is not always Metacompact | 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 $T$ be the Sorgenfrey line.
Let $T' = T \times T$ be Sorgenfrey's half-open square topology.
From Sorgenfrey Line is Metacompact, $T$ is a metacompact space.
From Sorgenfrey's Half-Open Square Topology is Not Metacompact, $T'$ is not a metacompact space.
Hence the result.
{{qed}} | 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 $T$ be the [[Definition:Sorgenfrey Line|Sorgenfrey line]].
Let $T' = T \times T$ be [[Definition:Sorgenfrey's Half-Open Square Topology|Sorgenfrey's half-open square topology]].
From [[Sorgenfrey Line is Metacompact]], $T$ is a [[Definition:Metacompact Space|metacompact space]].
From [[Sorgenfrey's Half-Open Squ... | Product of Metacompact Spaces is not always Metacompact | https://proofwiki.org/wiki/Product_of_Metacompact_Spaces_is_not_always_Metacompact | https://proofwiki.org/wiki/Product_of_Metacompact_Spaces_is_not_always_Metacompact | [
"Metacompact Spaces",
"Product Spaces"
] | [
"Definition:Indexing Set",
"Definition:Indexing Set/Family",
"Definition:Topological Space",
"Definition:Indexing Set/Family",
"Definition:Product Space (Topology)",
"Definition:Metacompact Space",
"Definition:Metacompact Space"
] | [
"Definition:Sorgenfrey Line",
"Definition:Sorgenfrey's Half-Open Square Topology",
"Sorgenfrey Line is Metacompact",
"Definition:Metacompact Space",
"Sorgenfrey's Half-Open Square Topology is Not Metacompact",
"Definition:Metacompact Space"
] |
proofwiki-13906 | Paracompactness is not always Preserved under Open Continuous Mapping | Let $T_A = \struct {X_A, \tau_A}$ be a topological space which is paracompact.
Let $T_B = \struct {X_B, \tau_B}$ be another topological space.
Let $\phi: T_A \to T_B$ be a mapping which is both continuous and open.
Then it is not necessarily the case that $T_B$ is also paracompact. | We have Open Continuous Image of Paracompact Space is not always Countably Metacompact.
We also have:
:Paracompact Space is Metacompact
:Metacompact Space is Countably Metacompact
Hence the result.
{{qed}} | Let $T_A = \struct {X_A, \tau_A}$ be a [[Definition:Topological Space|topological space]] which is [[Definition:Paracompact Space|paracompact]].
Let $T_B = \struct {X_B, \tau_B}$ be another [[Definition:Topological Space|topological space]].
Let $\phi: T_A \to T_B$ be a [[Definition:Mapping|mapping]] which is both [... | We have [[Open Continuous Image of Paracompact Space is not always Countably Metacompact]].
We also have:
:[[Paracompact Space is Metacompact]]
:[[Metacompact Space is Countably Metacompact]]
Hence the result.
{{qed}} | Paracompactness is not always Preserved under Open Continuous Mapping | https://proofwiki.org/wiki/Paracompactness_is_not_always_Preserved_under_Open_Continuous_Mapping | https://proofwiki.org/wiki/Paracompactness_is_not_always_Preserved_under_Open_Continuous_Mapping | [
"Paracompact Spaces",
"Continuous Mappings",
"Product Spaces",
"Open Mappings"
] | [
"Definition:Topological Space",
"Definition:Paracompact Space",
"Definition:Topological Space",
"Definition:Mapping",
"Definition:Continuous Mapping (Topology)/Everywhere",
"Definition:Open Mapping",
"Definition:Paracompact Space"
] | [
"Open Continuous Image of Paracompact Space is not always Countably Metacompact",
"Paracompact Space is Metacompact",
"Metacompact Space is Countably Metacompact"
] |
proofwiki-13907 | Union of Closure with Closure of Complement is Whole Space | Let $T = \struct {S, \tau}$ be a topological space.
Let $H \subseteq S$ be a subset of $S$.
Let $H^-$ denote the closure of $H$ in $T$.
Let $S \setminus H$ denote the complement of $H$ relative to $S$.
Then:
:$H^- \cup \paren {S \setminus H}^- = S$ | We have that:
:$H^- \cup \paren {S \setminus H}^- \subseteq S$
by definition of $S$.
From Union with Relative Complement:
:$H \cup \paren {S \setminus H} = S$
From Set is Subset of its Topological Closure:
{{begin-eqn}}
{{eqn | l = H
| o = \subseteq
| r = H^-
}}
{{eqn | l = S \setminus H
| o = \subset... | Let $T = \struct {S, \tau}$ be a [[Definition:Topological Space|topological space]].
Let $H \subseteq S$ be a [[Definition:Subset|subset]] of $S$.
Let $H^-$ denote the [[Definition:Closure (Topology)|closure]] of $H$ in $T$.
Let $S \setminus H$ denote the [[Definition:Relative Complement|complement of $H$ relative t... | We have that:
:$H^- \cup \paren {S \setminus H}^- \subseteq S$
by definition of $S$.
From [[Union with Relative Complement]]:
:$H \cup \paren {S \setminus H} = S$
From [[Set is Subset of its Topological Closure]]:
{{begin-eqn}}
{{eqn | l = H
| o = \subseteq
| r = H^-
}}
{{eqn | l = S \setminus H
|... | Union of Closure with Closure of Complement is Whole Space | https://proofwiki.org/wiki/Union_of_Closure_with_Closure_of_Complement_is_Whole_Space | https://proofwiki.org/wiki/Union_of_Closure_with_Closure_of_Complement_is_Whole_Space | [
"Set Closures",
"Set Union",
"Relative Complement"
] | [
"Definition:Topological Space",
"Definition:Subset",
"Definition:Closure (Topology)",
"Definition:Relative Complement"
] | [
"Union with Relative Complement",
"Set is Subset of its Topological Closure",
"Set Union Preserves Subsets",
"Definition:Set Equality",
"Category:Set Closures",
"Category:Set Union",
"Category:Relative Complement"
] |
proofwiki-13908 | Component of Point is not always Intersection of its Clopen Sets | Let $T = \struct {S, \tau}$ be a topological space.
Let $x \in S$.
Let $\map {\operatorname {Comp}_x} T$ denote the component of $x$ in $T$.
Let $K_x = \ds \bigcap_{x \mathop \in K} K$ clopen in $T$.
Then it is not always the case that $\map {\operatorname {Comp}_x} T = K_x$ | Note that from Clopen Set contains Components of All its Points:
:$\map {\operatorname {Comp}_x} T \subseteq K_x$
It remains to be demonstrated that it is not always the case that $K_x \subseteq \map {\operatorname {Comp}_x} T$.
Let $T$ be the nested rectangle space in the Euclidean plane.
Let $L_1$ and $L_2$ be the bo... | Let $T = \struct {S, \tau}$ be a [[Definition:Topological Space|topological space]].
Let $x \in S$.
Let $\map {\operatorname {Comp}_x} T$ denote the [[Definition:Component (Topology)|component of $x$ in $T$]].
Let $K_x = \ds \bigcap_{x \mathop \in K} K$ [[Definition:Clopen Set|clopen]] in $T$.
Then it is not alway... | Note that from [[Clopen Set contains Components of All its Points]]:
:$\map {\operatorname {Comp}_x} T \subseteq K_x$
It remains to be demonstrated that it is not always the case that $K_x \subseteq \map {\operatorname {Comp}_x} T$.
Let $T$ be the [[Definition:Nested Rectangle Topology|nested rectangle space]] in th... | Component of Point is not always Intersection of its Clopen Sets | https://proofwiki.org/wiki/Component_of_Point_is_not_always_Intersection_of_its_Clopen_Sets | https://proofwiki.org/wiki/Component_of_Point_is_not_always_Intersection_of_its_Clopen_Sets | [
"Clopen Sets",
"Components (Topology)"
] | [
"Definition:Topological Space",
"Definition:Component (Topology)",
"Definition:Clopen Set"
] | [
"Clopen Set contains Components of All its Points",
"Definition:Nested Rectangle Space",
"Definition:Euclidean Plane",
"Definition:Nested Rectangle Space/Boundary Line",
"Boundary Line in Nested Rectangle Space is Component",
"Definition:Component (Topology)",
"Union of Boundary Lines in Nested Rectangl... |
proofwiki-13909 | Complement of Clopen Set is Clopen | Let $T = \struct {S, \tau}$ be a topological space.
Let $H \subseteq S$ be a clopen set of $T$.
Let $\relcomp S H$ denote the complement of $H$ relative to $S$.
Then $\relcomp S H$ is also a clopen set of $T$. | By definition of clopen, $H$ is open in $T$.
By definition of closed set, $\relcomp S H$ is closed in $T$.
By definition of clopen, $H$ is closed in $T$.
By definition of closed set, $\relcomp S H$ is open in $T$.
Thus $\relcomp S H$ is both open in $T$ and closed in $T$.
Hence the result, by definition of clopen set.... | Let $T = \struct {S, \tau}$ be a [[Definition:Topological Space|topological space]].
Let $H \subseteq S$ be a [[Definition:Clopen Set|clopen set]] of $T$.
Let $\relcomp S H$ denote the [[Definition:Relative Complement|complement of $H$ relative to $S$]].
Then $\relcomp S H$ is also a [[Definition:Clopen Set|clopen ... | By definition of [[Definition:Clopen Set|clopen]], $H$ is [[Definition:Open Set (Topology)|open]] in $T$.
By definition of [[Definition:Closed Set (Topology)|closed set]], $\relcomp S H$ is [[Definition:Closed Set (Topology)|closed]] in $T$.
By definition of [[Definition:Clopen Set|clopen]], $H$ is [[Definition:Clos... | Complement of Clopen Set is Clopen | https://proofwiki.org/wiki/Complement_of_Clopen_Set_is_Clopen | https://proofwiki.org/wiki/Complement_of_Clopen_Set_is_Clopen | [
"Relative Complement",
"Clopen Sets"
] | [
"Definition:Topological Space",
"Definition:Clopen Set",
"Definition:Relative Complement",
"Definition:Clopen Set"
] | [
"Definition:Clopen Set",
"Definition:Open Set/Topology",
"Definition:Closed Set/Topology",
"Definition:Closed Set/Topology",
"Definition:Clopen Set",
"Definition:Closed Set/Topology",
"Definition:Closed Set/Topology",
"Definition:Open Set/Topology",
"Definition:Open Set/Topology",
"Definition:Clos... |
proofwiki-13910 | Clopen Set and Complement form Separation | Let $T = \struct {S, \tau}$ be a topological space.
Let $H \subseteq S$ be a clopen set of $T$.
Let $\relcomp S H$ be the complement of $H$ relative to $S$.
Then $H$ and $\relcomp S H$ form a separation of $T$. | By Set with Relative Complement forms Partition, $H$ and $\relcomp S H$ form a partition of $S$.
By Complement of Clopen Set is Clopen, $\relcomp S H$ is also a clopen set of $T$.
By definition of clopen set, both $H$ and $\relcomp S H$ are open in $T$.
Thus $H$ and $\relcomp S H$ are a pair of open sets in $T$ forming... | Let $T = \struct {S, \tau}$ be a [[Definition:Topological Space|topological space]].
Let $H \subseteq S$ be a [[Definition:Clopen Set|clopen set]] of $T$.
Let $\relcomp S H$ be the [[Definition:Relative Complement|complement of $H$ relative to $S$]].
Then $H$ and $\relcomp S H$ form a [[Definition:Separation (Topol... | By [[Set with Relative Complement forms Partition]], $H$ and $\relcomp S H$ form a [[Definition:Set Partition|partition]] of $S$.
By [[Complement of Clopen Set is Clopen]], $\relcomp S H$ is also a [[Definition:Clopen Set|clopen set]] of $T$.
By definition of [[Definition:Clopen Set|clopen set]], both $H$ and $\relco... | Clopen Set and Complement form Separation | https://proofwiki.org/wiki/Clopen_Set_and_Complement_form_Separation | https://proofwiki.org/wiki/Clopen_Set_and_Complement_form_Separation | [
"Clopen Sets",
"Separations"
] | [
"Definition:Topological Space",
"Definition:Clopen Set",
"Definition:Relative Complement",
"Definition:Separation (Topology)"
] | [
"Set Difference and Intersection form Partition/Corollary 2",
"Definition:Set Partition",
"Complement of Clopen Set is Clopen",
"Definition:Clopen Set",
"Definition:Clopen Set",
"Definition:Open Set/Topology",
"Definition:Open Set/Topology",
"Definition:Set Partition",
"Definition:Separation (Topolo... |
proofwiki-13911 | Path Component is not necessarily Injective Path Component | Let $T = \struct {S, \tau}$ be a topological space.
Let $P$ be a path component of $T$.
Then it is not necessarily the case that $P$ is also an injective path component of $T$. | Let $T = \struct {S, \tau_p}$ be a finite particular point space.
From Particular Point Space is Path-Connected, $T$ is path-connected.
Therefore $S$ is a path component in $T$.
But from Particular Point Space is not Injectively Path-Connected, $T$ is not injectively path-connected.
Therefore $S$ is not an injective pa... | Let $T = \struct {S, \tau}$ be a [[Definition:Topological Space|topological space]].
Let $P$ be a [[Definition:Path Component|path component]] of $T$.
Then it is not necessarily the case that $P$ is also an [[Definition:Injective Path Component|injective path component]] of $T$. | Let $T = \struct {S, \tau_p}$ be a [[Definition:Finite Particular Point Topology|finite particular point space]].
From [[Particular Point Space is Path-Connected]], $T$ is [[Definition:Path-Connected Space|path-connected]].
Therefore $S$ is a [[Definition:Path Component|path component]] in $T$.
But from [[Particular... | Path Component is not necessarily Injective Path Component | https://proofwiki.org/wiki/Path_Component_is_not_necessarily_Injective_Path_Component | https://proofwiki.org/wiki/Path_Component_is_not_necessarily_Injective_Path_Component | [
"Injective Path Components",
"Path Components"
] | [
"Definition:Topological Space",
"Definition:Path Component",
"Definition:Injective Path Component"
] | [
"Definition:Particular Point Topology/Finite",
"Particular Point Space is Path-Connected",
"Definition:Path-Connected/Topological Space",
"Definition:Path Component",
"Particular Point Space is not Injectively Path-Connected",
"Definition:Injectively Path-Connected/Topological Space",
"Definition:Inject... |
proofwiki-13912 | Component is not necessarily Path Component | Let $T = \struct {S, \tau}$ be a topological space.
Let $C$ be a component of $T$.
Then it is not necessarily the case that $C$ is also an path component of $T$. | Let $C$ be the closed topologist's sine curve embedded in the real Euclidean plane.
From Closed Topologist's Sine Curve is Connected, $C$ is connected in $T$
Therefore $C$ is a component in the subspace of $T$ induced by $C$.
From Closed Topologist's Sine Curve is not Path-Connected, $C$ is not path-connected.
Therefor... | Let $T = \struct {S, \tau}$ be a [[Definition:Topological Space|topological space]].
Let $C$ be a [[Definition:Component (Topology)|component]] of $T$.
Then it is not necessarily the case that $C$ is also an [[Definition:Path Component|path component]] of $T$. | Let $C$ be the [[Definition:Closed Topologist's Sine Curve|closed topologist's sine curve]] embedded in the [[Definition:Real Euclidean Space|real Euclidean plane]].
From [[Closed Topologist's Sine Curve is Connected]], $C$ is [[Definition:Connected Set (Topology)|connected]] in $T$
Therefore $C$ is a [[Definition:Co... | Component is not necessarily Path Component | https://proofwiki.org/wiki/Component_is_not_necessarily_Path_Component | https://proofwiki.org/wiki/Component_is_not_necessarily_Path_Component | [
"Components (Topology)",
"Path Components"
] | [
"Definition:Topological Space",
"Definition:Component (Topology)",
"Definition:Path Component"
] | [
"Definition:Closed Topologist's Sine Curve",
"Definition:Euclidean Space/Real",
"Closed Topologist's Sine Curve is Connected",
"Definition:Connected Set (Topology)",
"Definition:Component (Topology)",
"Definition:Topological Subspace",
"Closed Topologist's Sine Curve is not Path-Connected",
"Definitio... |
proofwiki-13913 | Quasicomponent is not necessarily Component | Let $T = \struct {S, \tau}$ be a topological space.
Let $Q$ be a quasicomponent of $T$.
Then it is not necessarily the case that $C$ is also a component of $T$. | From Component of Point is not always Intersection of its Clopen Sets, the set intersection of the clopen sets containing a point $x$ may not always be contained in the component of $x$.
The result follows from Quasicomponent is Intersection of Clopen Sets.
{{qed}} | Let $T = \struct {S, \tau}$ be a [[Definition:Topological Space|topological space]].
Let $Q$ be a [[Definition:Quasicomponent|quasicomponent]] of $T$.
Then it is not necessarily the case that $C$ is also a [[Definition:Component (Topology)|component]] of $T$. | From [[Component of Point is not always Intersection of its Clopen Sets]], the [[Definition:Set Intersection|set intersection]] of the [[Definition:Clopen Set|clopen sets]] containing a point $x$ may not always be [[Definition:Subset|contained]] in the [[Definition:Component (Topology)|component]] of $x$.
The result f... | Quasicomponent is not necessarily Component | https://proofwiki.org/wiki/Quasicomponent_is_not_necessarily_Component | https://proofwiki.org/wiki/Quasicomponent_is_not_necessarily_Component | [
"Components (Topology)",
"Quasicomponents"
] | [
"Definition:Topological Space",
"Definition:Quasicomponent",
"Definition:Component (Topology)"
] | [
"Component of Point is not always Intersection of its Clopen Sets",
"Definition:Set Intersection",
"Definition:Clopen Set",
"Definition:Subset",
"Definition:Component (Topology)",
"Quasicomponent is Intersection of Clopen Sets"
] |
proofwiki-13914 | Simple Infinite Continued Fraction is Uniquely Determined by Limit | Let $\sequence {a_n}_{n \mathop \ge 0}$ and $\sequence {b_n}_{n \mathop \ge 0}$ be simple infinite continued fractions in $\R$.
Let $\sequence {a_n}_{n \mathop \ge 0}$ and $\sequence {b_n}_{n \mathop \ge 0}$ have the same limit.
Then they are equal. | Follows immediately from Continued Fraction Expansion of Limit of Simple Infinite Continued Fraction equals Expansion Itself.
{{qed}} | Let $\sequence {a_n}_{n \mathop \ge 0}$ and $\sequence {b_n}_{n \mathop \ge 0}$ be [[Definition:Simple Infinite Continued Fraction|simple infinite continued fractions]] in $\R$.
Let $\sequence {a_n}_{n \mathop \ge 0}$ and $\sequence {b_n}_{n \mathop \ge 0}$ have the same [[Definition:Limit of Continued Fraction|limit]... | Follows immediately from [[Continued Fraction Expansion of Limit of Simple Infinite Continued Fraction equals Expansion Itself]].
{{qed}} | Simple Infinite Continued Fraction is Uniquely Determined by Limit/Proof 1 | https://proofwiki.org/wiki/Simple_Infinite_Continued_Fraction_is_Uniquely_Determined_by_Limit | https://proofwiki.org/wiki/Simple_Infinite_Continued_Fraction_is_Uniquely_Determined_by_Limit/Proof_1 | [
"Simple Infinite Continued Fraction is Uniquely Determined by Limit",
"Simple Continued Fractions"
] | [
"Definition:Simple Continued Fraction/Infinite",
"Definition:Value of Continued Fraction/Infinite"
] | [
"Continued Fraction Expansion of Limit of Simple Infinite Continued Fraction equals Expansion Itself"
] |
proofwiki-13915 | Simple Infinite Continued Fraction is Uniquely Determined by Limit | Let $\sequence {a_n}_{n \mathop \ge 0}$ and $\sequence {b_n}_{n \mathop \ge 0}$ be simple infinite continued fractions in $\R$.
Let $\sequence {a_n}_{n \mathop \ge 0}$ and $\sequence {b_n}_{n \mathop \ge 0}$ have the same limit.
Then they are equal. | Recall that by Simple Infinite Continued Fraction Converges, they do indeed have a limit.
The result will be achieved by the Second Principle of Mathematical Induction.
Suppose $\sqbrk {a_0, a_1, a_2, \ldots} = \sqbrk {b_0, b_1, b_2, \ldots}$ have the same value.
First we note that if $\sqbrk {a_0, a_1, a_2, \ldots} = ... | Let $\sequence {a_n}_{n \mathop \ge 0}$ and $\sequence {b_n}_{n \mathop \ge 0}$ be [[Definition:Simple Infinite Continued Fraction|simple infinite continued fractions]] in $\R$.
Let $\sequence {a_n}_{n \mathop \ge 0}$ and $\sequence {b_n}_{n \mathop \ge 0}$ have the same [[Definition:Limit of Continued Fraction|limit]... | Recall that by [[Simple Infinite Continued Fraction Converges]], they do indeed have a [[Definition:Limit of Continued Fraction|limit]].
The result will be achieved by the [[Second Principle of Mathematical Induction]].
Suppose $\sqbrk {a_0, a_1, a_2, \ldots} = \sqbrk {b_0, b_1, b_2, \ldots}$ have the same value.
F... | Simple Infinite Continued Fraction is Uniquely Determined by Limit/Proof 2 | https://proofwiki.org/wiki/Simple_Infinite_Continued_Fraction_is_Uniquely_Determined_by_Limit | https://proofwiki.org/wiki/Simple_Infinite_Continued_Fraction_is_Uniquely_Determined_by_Limit/Proof_2 | [
"Simple Infinite Continued Fraction is Uniquely Determined by Limit",
"Simple Continued Fractions"
] | [
"Definition:Simple Continued Fraction/Infinite",
"Definition:Value of Continued Fraction/Infinite"
] | [
"Simple Infinite Continued Fraction Converges",
"Definition:Value of Continued Fraction/Infinite",
"Second Principle of Mathematical Induction",
"Definition:Partial Denominator",
"Definition:Simple Continued Fraction/Infinite"
] |
proofwiki-13916 | Continued Fraction Expansion of Irrational Number Converges to Number Itself | Let $x$ be an irrational number.
Then the continued fraction expansion of $x$ converges to $x$. | Let $\sequence {a_0, a_1, \ldots}$ be its continued fraction expansion.
Let $\sequence {p_n}_{n \mathop \ge 0}$ and $\sequence {q_n}_{n \mathop \ge 0}$ be its numerators and denominators.
Then $C_n = p_n / q_n$ is the $n$th convergent.
By Accuracy of Convergents of Continued Fraction Expansion of Irrational Number, for... | Let $x$ be an [[Definition:Irrational Number|irrational number]].
Then the [[Definition:Continued Fraction Expansion of Irrational Number|continued fraction expansion]] of $x$ [[Definition:Convergent Continued Fraction|converges]] to $x$. | Let $\sequence {a_0, a_1, \ldots}$ be its [[Definition:Continued Fraction Expansion of Irrational Number|continued fraction expansion]].
Let $\sequence {p_n}_{n \mathop \ge 0}$ and $\sequence {q_n}_{n \mathop \ge 0}$ be its [[Definition:Numerators and Denominators of Continued Fraction|numerators and denominators]].
... | Continued Fraction Expansion of Irrational Number Converges to Number Itself | https://proofwiki.org/wiki/Continued_Fraction_Expansion_of_Irrational_Number_Converges_to_Number_Itself | https://proofwiki.org/wiki/Continued_Fraction_Expansion_of_Irrational_Number_Converges_to_Number_Itself | [
"Continued Fractions"
] | [
"Definition:Irrational Number",
"Definition:Continued Fraction Expansion/Real Number",
"Definition:Convergent Continued Fraction"
] | [
"Definition:Continued Fraction Expansion/Real Number",
"Definition:Numerators and Denominators of Continued Fraction",
"Definition:Convergent of Continued Fraction",
"Accuracy of Convergents of Continued Fraction Expansion of Irrational Number",
"Lower Bounds for Denominators of Simple Continued Fraction",
... |
proofwiki-13917 | Continued Fraction Expansion of Limit of Simple Infinite Continued Fraction equals Expansion Itself | Let $\sequence {a_n}_{n \mathop \ge 0}$ be a simple infinite continued fractions in $\R$.
Then $\sequence {a_n}_{n \mathop \ge 0}$ converges to an irrational number, whose continued fraction expansion is $\sequence {a_n}_{n \mathop \ge 0}$. | By Simple Infinite Continued Fraction Converges to Irrational Number, the value of $\sequence {a_n}_{n \mathop \ge 0}$ exists and is irrational.
Let $\sequence {b_n}_{n \mathop \ge 0}$ be its continued fraction expansion.
By Continued Fraction Expansion of Irrational Number Converges to Number Itself, $\sequence {a_n}_... | Let $\sequence {a_n}_{n \mathop \ge 0}$ be a [[Definition:Simple Infinite Continued Fraction|simple infinite continued fractions]] in $\R$.
Then $\sequence {a_n}_{n \mathop \ge 0}$ [[Definition:Convergent Continued Fraction|converges]] to an [[Definition:Irrational Number|irrational number]], whose [[Definition:Conti... | By [[Simple Infinite Continued Fraction Converges to Irrational Number]], the [[Definition:Value of Infinite Continued Fraction|value]] of $\sequence {a_n}_{n \mathop \ge 0}$ exists and is [[Definition:Irrational Number|irrational]].
Let $\sequence {b_n}_{n \mathop \ge 0}$ be its [[Definition:Continued Fraction Expans... | Continued Fraction Expansion of Limit of Simple Infinite Continued Fraction equals Expansion Itself | https://proofwiki.org/wiki/Continued_Fraction_Expansion_of_Limit_of_Simple_Infinite_Continued_Fraction_equals_Expansion_Itself | https://proofwiki.org/wiki/Continued_Fraction_Expansion_of_Limit_of_Simple_Infinite_Continued_Fraction_equals_Expansion_Itself | [
"Simple Continued Fractions"
] | [
"Definition:Simple Continued Fraction/Infinite",
"Definition:Convergent Continued Fraction",
"Definition:Irrational Number",
"Definition:Continued Fraction Expansion/Real Number"
] | [
"Simple Infinite Continued Fraction Converges to Irrational Number",
"Definition:Value of Continued Fraction/Infinite",
"Definition:Irrational Number",
"Definition:Continued Fraction Expansion/Real Number",
"Continued Fraction Expansion of Irrational Number Converges to Number Itself",
"Definition:Value o... |
proofwiki-13918 | Correspondence between Irrational Numbers and Simple Infinite Continued Fractions | Let $\R \setminus \Q$ be the set of irrational numbers.
Let $S$ be the set of all simple infinite continued fractions in $\R$.
The mappings:
:$\R \setminus \Q \to S$ that sends an irrational number to its continued fraction expansion
:$S \to \R \setminus \Q$ that sends a simple infinite continued fractions to its value... | Note that indeed a Simple Infinite Continued Fraction Converges to Irrational Number.
The result follows from:
: Continued Fraction Expansion of Irrational Number Converges to Number Itself
: Continued Fraction Expansion of Limit of Simple Infinite Continued Fraction equals Expansion Itself
{{qed}} | Let $\R \setminus \Q$ be the [[Definition:Set|set]] of [[Definition:Irrational Number|irrational numbers]].
Let $S$ be the [[Definition:Set|set]] of all [[Definition:Simple Infinite Continued Fraction|simple infinite continued fractions]] in $\R$.
The [[Definition:Mapping|mappings]]:
:$\R \setminus \Q \to S$ that se... | Note that indeed a [[Simple Infinite Continued Fraction Converges to Irrational Number]].
The result follows from:
: [[Continued Fraction Expansion of Irrational Number Converges to Number Itself]]
: [[Continued Fraction Expansion of Limit of Simple Infinite Continued Fraction equals Expansion Itself]]
{{qed}} | Correspondence between Irrational Numbers and Simple Infinite Continued Fractions | https://proofwiki.org/wiki/Correspondence_between_Irrational_Numbers_and_Simple_Infinite_Continued_Fractions | https://proofwiki.org/wiki/Correspondence_between_Irrational_Numbers_and_Simple_Infinite_Continued_Fractions | [
"Simple Continued Fractions"
] | [
"Definition:Set",
"Definition:Irrational Number",
"Definition:Set",
"Definition:Simple Continued Fraction/Infinite",
"Definition:Mapping",
"Definition:Irrational Number",
"Definition:Continued Fraction Expansion/Real Number",
"Definition:Simple Continued Fraction/Infinite",
"Definition:Value of Cont... | [
"Simple Infinite Continued Fraction Converges to Irrational Number",
"Continued Fraction Expansion of Irrational Number Converges to Number Itself",
"Continued Fraction Expansion of Limit of Simple Infinite Continued Fraction equals Expansion Itself"
] |
proofwiki-13919 | Ultraconnected Space is Connected | Let $T = \struct {S, \tau}$ be a topological space which is ultraconnected.
Then $T$ is connected. | Let $T = \struct {S, \tau}$ be a topological space which is ultraconnected.
From Ultraconnected Space is Path-Connected, $T$ is path-connected.
The result follows from Path-Connected Space is Connected.
{{qed}} | Let $T = \struct {S, \tau}$ be a [[Definition:Topological Space|topological space]] which is [[Definition:Ultraconnected Space|ultraconnected]].
Then $T$ is [[Definition:Connected Topological Space|connected]]. | Let $T = \struct {S, \tau}$ be a [[Definition:Topological Space|topological space]] which is [[Definition:Ultraconnected Space|ultraconnected]].
From [[Ultraconnected Space is Path-Connected]], $T$ is [[Definition:Path-Connected Space|path-connected]].
The result follows from [[Path-Connected Space is Connected]].
{{... | Ultraconnected Space is Connected | https://proofwiki.org/wiki/Ultraconnected_Space_is_Connected | https://proofwiki.org/wiki/Ultraconnected_Space_is_Connected | [
"Ultraconnected Spaces",
"Connected Topological Spaces",
"Sequence of Implications of Connectedness Properties"
] | [
"Definition:Topological Space",
"Definition:Ultraconnected Space",
"Definition:Connected Topological Space"
] | [
"Definition:Topological Space",
"Definition:Ultraconnected Space",
"Ultraconnected Space is Path-Connected",
"Definition:Path-Connected/Topological Space",
"Path-Connected Space is Connected"
] |
proofwiki-13920 | Irreducible Space is not necessarily Path-Connected | Let $T = \struct {S, \tau}$ be a topological space which is irreducible.
Then $T$ is not necessarily path-connected. | Let $T$ be a countable finite complement space.
From Finite Complement Space is Irreducible, $T$ is an irreducible space.
From Countable Finite Complement Space is not Path-Connected, $T$ is not path-connected.
Hence the result.
{{qed}} | Let $T = \struct {S, \tau}$ be a [[Definition:Topological Space|topological space]] which is [[Definition:Irreducible Space|irreducible]].
Then $T$ is not necessarily [[Definition:Path-Connected Space|path-connected]]. | Let $T$ be a [[Definition:Countable Finite Complement Topology|countable finite complement space]].
From [[Finite Complement Space is Irreducible]], $T$ is an [[Definition:Irreducible Space|irreducible space]].
From [[Countable Finite Complement Space is not Path-Connected]], $T$ is not [[Definition:Path-Connected Sp... | Irreducible Space is not necessarily Path-Connected | https://proofwiki.org/wiki/Irreducible_Space_is_not_necessarily_Path-Connected | https://proofwiki.org/wiki/Irreducible_Space_is_not_necessarily_Path-Connected | [
"Irreducible Spaces",
"Path-Connected Spaces",
"Sequence of Implications of Connectedness Properties"
] | [
"Definition:Topological Space",
"Definition:Irreducible Space",
"Definition:Path-Connected/Topological Space"
] | [
"Definition:Finite Complement Topology/Countable",
"Finite Complement Space is Irreducible",
"Definition:Irreducible Space",
"Countable Finite Complement Space is not Path-Connected",
"Definition:Path-Connected/Topological Space"
] |
proofwiki-13921 | Ultraconnected Space is not necessarily Injectively Path-Connected | Let $T = \struct {S, \tau}$ be a topological space which is ultraconnected.
Then $T$ is not necessarily injectively path-connected. | Let $T$ be an excluded point space.
From Excluded Point Space is Ultraconnected, $T$ is an ultraconnected space.
From Excluded Point Space is not Injectively Path-Connected, $T$ is not injectively path-connected.
Hence the result.
{{qed}} | Let $T = \struct {S, \tau}$ be a [[Definition:Topological Space|topological space]] which is [[Definition:Ultraconnected Space|ultraconnected]].
Then $T$ is not necessarily [[Definition:Injectively Path-Connected Space|injectively path-connected]]. | Let $T$ be an [[Definition:Excluded Point Topology|excluded point space]].
From [[Excluded Point Space is Ultraconnected]], $T$ is an [[Definition:Ultraconnected Space|ultraconnected space]].
From [[Excluded Point Space is not Injectively Path-Connected]], $T$ is not [[Definition:Injectively Path-Connected Space|inje... | Ultraconnected Space is not necessarily Injectively Path-Connected | https://proofwiki.org/wiki/Ultraconnected_Space_is_not_necessarily_Injectively_Path-Connected | https://proofwiki.org/wiki/Ultraconnected_Space_is_not_necessarily_Injectively_Path-Connected | [
"Ultraconnected Spaces",
"Injectively Path-Connected Spaces",
"Sequence of Implications of Connectedness Properties"
] | [
"Definition:Topological Space",
"Definition:Ultraconnected Space",
"Definition:Injectively Path-Connected/Topological Space"
] | [
"Definition:Excluded Point Topology",
"Excluded Point Space is Ultraconnected",
"Definition:Ultraconnected Space",
"Excluded Point Space is not Injectively Path-Connected",
"Definition:Injectively Path-Connected/Topological Space"
] |
proofwiki-13922 | Correspondence between Rational Numbers and Simple Finite Continued Fractions | Let $\Q$ be the set of rational numbers.
Let $S$ be the set of all simple finite continued fractions in $\Q$, whose last partial denominators is not $1$.
The mappings:
:$\Q \to S$ that sends an rational number to its continued fraction expansion
:$S \to \Q$ that sends a simple finite continued fractions to its value
ar... | Note that indeed Simple Finite Continued Fraction has Rational Value.
The result follows from:
:Value of Continued Fraction Expansion of Rational Number equals Number Itself
:Continued Fraction Expansion of Value of Simple Finite Continued Fraction equals Expansion Itself
{{qed}} | Let $\Q$ be the [[Definition:Set|set]] of [[Definition:Rational Number|rational numbers]].
Let $S$ be the [[Definition:Set|set]] of all [[Definition:Simple Finite Continued Fraction|simple finite continued fractions]] in $\Q$, whose last [[Definition:Partial Denominator|partial denominators]] is not $1$.
The [[Defin... | Note that indeed [[Simple Finite Continued Fraction has Rational Value]].
The result follows from:
:[[Value of Continued Fraction Expansion of Rational Number equals Number Itself]]
:[[Continued Fraction Expansion of Value of Simple Finite Continued Fraction equals Expansion Itself]]
{{qed}} | Correspondence between Rational Numbers and Simple Finite Continued Fractions | https://proofwiki.org/wiki/Correspondence_between_Rational_Numbers_and_Simple_Finite_Continued_Fractions | https://proofwiki.org/wiki/Correspondence_between_Rational_Numbers_and_Simple_Finite_Continued_Fractions | [
"Simple Continued Fractions",
"Rational Numbers"
] | [
"Definition:Set",
"Definition:Rational Number",
"Definition:Set",
"Definition:Simple Continued Fraction/Finite",
"Definition:Partial Denominator",
"Definition:Mapping",
"Definition:rational Number",
"Definition:Continued Fraction Expansion/Real Number",
"Definition:Simple Continued Fraction/Finite",... | [
"Simple Finite Continued Fraction has Rational Value",
"Value of Continued Fraction Expansion of Rational Number equals Number Itself",
"Continued Fraction Expansion of Value of Simple Finite Continued Fraction equals Expansion Itself"
] |
proofwiki-13923 | Existence of Non-Locally Connected Space where Components and Quasicomponents are Equal | There exists at least one example of a topological space which is not locally connected, but whose components and quasicomponents are equal. | Let $T$ be the Arens-Fort space.
From Arens-Fort Space is not Locally Connected, $T$ is not a locally connected space.
The result follows from Components and Quasicomponents of Arens-Fort Space are Equal.
{{qed}} | There exists at least one example of a [[Definition:Topological Space|topological space]] which is not [[Definition:Locally Connected Space|locally connected]], but whose [[Definition:Component (Topology)|components]] and [[Definition:Quasicomponent|quasicomponents]] are equal. | Let $T$ be the [[Definition:Arens-Fort Space|Arens-Fort space]].
From [[Arens-Fort Space is not Locally Connected]], $T$ is not a [[Definition:Locally Connected Space|locally connected space]].
The result follows from [[Components and Quasicomponents of Arens-Fort Space are Equal]].
{{qed}} | Existence of Non-Locally Connected Space where Components and Quasicomponents are Equal | https://proofwiki.org/wiki/Existence_of_Non-Locally_Connected_Space_where_Components_and_Quasicomponents_are_Equal | https://proofwiki.org/wiki/Existence_of_Non-Locally_Connected_Space_where_Components_and_Quasicomponents_are_Equal | [
"Locally Connected Spaces",
"Components (Topology)",
"Quasicomponents"
] | [
"Definition:Topological Space",
"Definition:Locally Connected Space",
"Definition:Component (Topology)",
"Definition:Quasicomponent"
] | [
"Definition:Arens-Fort Space",
"Arens-Fort Space is not Locally Connected",
"Definition:Locally Connected Space",
"Components and Quasicomponents of Arens-Fort Space are Equal"
] |
proofwiki-13924 | Finite Irreducible Space is Path-Connected | Let $T = \struct {S, \tau}$ be a finite irreducible topological space.
Then $T$ is path-connected. | By Power Set of Finite Set is Finite, the power set $\powerset S$ is finite.
By Subset of Finite Set is Finite, $\tau \subseteq \powerset S$ is finite.
The result follows from Irreducible Space with Finitely Many Open Sets is Path-Connected.
{{qed}} | Let $T = \struct {S, \tau}$ be a [[Definition:Finite Set|finite]] [[Definition:Irreducible Space|irreducible topological space]].
Then $T$ is [[Definition:Path-Connected Space|path-connected]]. | By [[Power Set of Finite Set is Finite]], the [[Definition:Power Set|power set]] $\powerset S$ is [[Definition:Finite Set|finite]].
By [[Subset of Finite Set is Finite]], $\tau \subseteq \powerset S$ is [[Definition:Finite Set|finite]].
The result follows from [[Irreducible Space with Finitely Many Open Sets is Path-... | Finite Irreducible Space is Path-Connected | https://proofwiki.org/wiki/Finite_Irreducible_Space_is_Path-Connected | https://proofwiki.org/wiki/Finite_Irreducible_Space_is_Path-Connected | [
"Irreducible Spaces",
"Path-Connected Spaces",
"Finite Topological Spaces",
"Sequence of Implications of Connectedness Properties"
] | [
"Definition:Finite Set",
"Definition:Irreducible Space",
"Definition:Path-Connected/Topological Space"
] | [
"Power Set of Finite Set is Finite",
"Definition:Power Set",
"Definition:Finite Set",
"Subset of Finite Set is Finite",
"Definition:Finite Set",
"Irreducible Space with Finitely Many Open Sets is Path-Connected"
] |
proofwiki-13925 | Locally Connected Space is not necessarily Connected | Let $T = \struct {S, \tau}$ be a topological space which is locally connected.
Then it is not necessarily the case that $T$ is also a connected space. | Let $T$ be a discrete topological space with more than $1$ point.
From Discrete Space is Locally Connected, $T$ is a locally connected space.
From Non-Trivial Discrete Space is not Connected, $T$ is not a connected space.
Hence the result.
{{qed}} | Let $T = \struct {S, \tau}$ be a [[Definition:Topological Space|topological space]] which is [[Definition:Locally Connected Space|locally connected]].
Then it is not necessarily the case that $T$ is also a [[Definition:Connected Topological Space|connected space]]. | Let $T$ be a [[Definition:Discrete Topology|discrete topological space]] with more than $1$ point.
From [[Discrete Space is Locally Connected]], $T$ is a [[Definition:Locally Connected Space|locally connected space]].
From [[Non-Trivial Discrete Space is not Connected]], $T$ is not a [[Definition:Connected Topologica... | Locally Connected Space is not necessarily Connected | https://proofwiki.org/wiki/Locally_Connected_Space_is_not_necessarily_Connected | https://proofwiki.org/wiki/Locally_Connected_Space_is_not_necessarily_Connected | [
"Locally Connected Spaces",
"Connected Topological Spaces"
] | [
"Definition:Topological Space",
"Definition:Locally Connected Space",
"Definition:Connected Topological Space"
] | [
"Definition:Discrete Topology",
"Discrete Space is Locally Connected",
"Definition:Locally Connected Space",
"Non-Trivial Discrete Space is not Connected",
"Definition:Connected Topological Space"
] |
proofwiki-13926 | Connected Space is not necessarily Locally Connected | Let $T = \struct {S, \tau}$ be a topological space which is connected.
Then it is not necessarily the case that $T$ is also a locally connected space. | Let $C$ be the closed topologist's sine curve embedded in the real Euclidean plane.
From Closed Topologist's Sine Curve is Connected, $C$ is connected in $T$
From Closed Topologist's Sine Curve is not Locally Connected, $C$ is not locally connected.
Hence the result.
{{qed}} | Let $T = \struct {S, \tau}$ be a [[Definition:Topological Space|topological space]] which is [[Definition:Connected Topological Space|connected]].
Then it is not necessarily the case that $T$ is also a [[Definition:Locally Connected Space|locally connected space]]. | Let $C$ be the [[Definition:Closed Topologist's Sine Curve|closed topologist's sine curve]] embedded in the [[Definition:Real Euclidean Space|real Euclidean plane]].
From [[Closed Topologist's Sine Curve is Connected]], $C$ is [[Definition:Connected Set (Topology)|connected]] in $T$
From [[Closed Topologist's Sine Cu... | Connected Space is not necessarily Locally Connected | https://proofwiki.org/wiki/Connected_Space_is_not_necessarily_Locally_Connected | https://proofwiki.org/wiki/Connected_Space_is_not_necessarily_Locally_Connected | [
"Locally Connected Spaces",
"Connected Topological Spaces",
"Sequence of Implications of Connectedness Properties"
] | [
"Definition:Topological Space",
"Definition:Connected Topological Space",
"Definition:Locally Connected Space"
] | [
"Definition:Closed Topologist's Sine Curve",
"Definition:Euclidean Space/Real",
"Closed Topologist's Sine Curve is Connected",
"Definition:Connected Set (Topology)",
"Closed Topologist's Sine Curve is not Locally Connected",
"Definition:Locally Connected Space"
] |
proofwiki-13927 | Irreducible Space with Finitely Many Open Sets is Path-Connected | Let $T = \struct {S, \tau}$ be an irreducible topological space.
Let its topology $\tau$ be finite.
Then $T$ is path-connected. | Follows immediately from:
* Irreducible Space with Finitely Many Open Sets has Generic Point
* Topological Space with Generic Point is Path-Connected
{{qed}} | Let $T = \struct {S, \tau}$ be an [[Definition:Irreducible Space|irreducible topological space]].
Let its [[Definition:Topology|topology]] $\tau$ be [[Definition:Finite Set|finite]].
Then $T$ is [[Definition:Path-Connected Space|path-connected]]. | Follows immediately from:
* [[Irreducible Space with Finitely Many Open Sets has Generic Point]]
* [[Topological Space with Generic Point is Path-Connected]]
{{qed}} | Irreducible Space with Finitely Many Open Sets is Path-Connected | https://proofwiki.org/wiki/Irreducible_Space_with_Finitely_Many_Open_Sets_is_Path-Connected | https://proofwiki.org/wiki/Irreducible_Space_with_Finitely_Many_Open_Sets_is_Path-Connected | [
"Irreducible Spaces",
"Path-Connected Spaces",
"Sequence of Implications of Connectedness Properties"
] | [
"Definition:Irreducible Space",
"Definition:Topology",
"Definition:Finite Set",
"Definition:Path-Connected/Topological Space"
] | [
"Irreducible Space with Finitely Many Open Sets has Generic Point",
"Topological Space with Generic Point is Path-Connected"
] |
proofwiki-13928 | Topological Space with Generic Point is Path-Connected | Let $T = \struct {S, \tau}$ be a topological space.
Let $T$ have a generic point $g \in S$.
Then $T$ is path-connected. | By Path-Connectedness is Equivalence Relation, it suffices to prove that every point is path-connected with $g$.
Let $x \in S$.
Define a path $\gamma: \closedint 0 1 \to S$ by:
:<nowiki>$\map \gamma t = \begin{cases}
x & : t \le \dfrac 1 2 \\
g & : t > \dfrac 1 2
\end{cases}$</nowiki>
We show that $\gamma$ is indeed co... | Let $T = \struct {S, \tau}$ be a [[Definition:Topological Space|topological space]].
Let $T$ have a [[Definition:Generic Point of Topological Space|generic point]] $g \in S$.
Then $T$ is [[Definition:Path-Connected Space|path-connected]]. | By [[Path-Connectedness is Equivalence Relation]], it suffices to prove that every [[Definition:Point of Set|point]] is [[Definition:Path-Connected Points|path-connected]] with $g$.
Let $x \in S$.
Define a [[Definition:Path (Topology)|path]] $\gamma: \closedint 0 1 \to S$ by:
:<nowiki>$\map \gamma t = \begin{cases}
x... | Topological Space with Generic Point is Path-Connected | https://proofwiki.org/wiki/Topological_Space_with_Generic_Point_is_Path-Connected | https://proofwiki.org/wiki/Topological_Space_with_Generic_Point_is_Path-Connected | [
"Path-Connected Spaces"
] | [
"Definition:Topological Space",
"Definition:Generic Point of Topological Space",
"Definition:Path-Connected/Topological Space"
] | [
"Path-Connectedness is Equivalence Relation",
"Definition:Element",
"Definition:Path-Connected/Points",
"Definition:Path (Topology)",
"Definition:Continuous Mapping (Topology)/Everywhere",
"Definition:Open Set/Topology",
"Definition:Non-Empty Set",
"Definition:Generic Point of Topological Space",
"D... |
proofwiki-13929 | Floor of Simple Finite Continued Fraction | Let $\sequence {a_k}_{k \mathop \ge 0}$ be a simple finite continued fraction of length $n \ge 0$.
Let $x = \sqbrk {a_0, \ldots, a_n}$ be its value.
Then the floor of $x$ is the partial denominator $a_0$:
:$\floor x = a_0$
unless $n = 1$ and $a_1 = 1$, in which case:
:$\floor x = x = a_0 + 1$ | ;Length $0$
Let $n = 0$.
Then:
{{begin-eqn}}
{{eqn | l = x
| r = \sqbrk {a_0}
| c = {{hypothesis}}: $n = 0$
}}
{{eqn | r = a_0
| c = {{Defof|Value of Finite Continued Fraction}}
}}
{{eqn | ll= \leadsto
| l = \floor x
| r = \floor {a_0}
| c =
}}
{{eqn | r = a_0
| c = Real Numbe... | Let $\sequence {a_k}_{k \mathop \ge 0}$ be a [[Definition:Simple Finite Continued Fraction|simple finite continued fraction]] of [[Definition:Length of Continued Fraction|length]] $n \ge 0$.
Let $x = \sqbrk {a_0, \ldots, a_n}$ be its [[Definition:Value of Finite Continued Fraction|value]].
Then the [[Definition:Floo... | ;Length $0$
Let $n = 0$.
Then:
{{begin-eqn}}
{{eqn | l = x
| r = \sqbrk {a_0}
| c = {{hypothesis}}: $n = 0$
}}
{{eqn | r = a_0
| c = {{Defof|Value of Finite Continued Fraction}}
}}
{{eqn | ll= \leadsto
| l = \floor x
| r = \floor {a_0}
| c =
}}
{{eqn | r = a_0
| c = [[Real ... | Floor of Simple Finite Continued Fraction | https://proofwiki.org/wiki/Floor_of_Simple_Finite_Continued_Fraction | https://proofwiki.org/wiki/Floor_of_Simple_Finite_Continued_Fraction | [
"Simple Continued Fractions",
"Floor Function"
] | [
"Definition:Simple Continued Fraction/Finite",
"Definition:Length of Continued Fraction",
"Definition:Value of Continued Fraction/Finite",
"Definition:Floor Function",
"Definition:Partial Denominator"
] | [
"Real Number is Integer iff equals Floor",
"Real Number is Integer iff equals Floor",
"Real Number is Integer iff equals Floor",
"Value of Finite Continued Fraction of Real Numbers is at Least First Term",
"Value of Finite Continued Fraction of Real Numbers is at Least First Term"
] |
proofwiki-13930 | Value of Finite Continued Fraction of Strictly Positive Real Numbers is Strictly Positive | Let $\sequence {a_0, \ldots, a_n}$ be a finite continued fraction in $\R$ of length $n \ge 0$.
Let all partial denominators $a_k > 0$ be strictly positive.
Let $x = \sqbrk {a_0, a_1, \ldots, a_n}$ be its value.
Then $x > 0$. | {{proof wanted|use Definition:Value of Continued Fraction}} | Let $\sequence {a_0, \ldots, a_n}$ be a [[Definition:Finite Continued Fraction|finite continued fraction]] in $\R$ of [[Definition:Length of Continued Fraction|length]] $n \ge 0$.
Let all [[Definition:Partial Denominator|partial denominators]] $a_k > 0$ be [[Definition:Strictly Positive Real Number|strictly positive]]... | {{proof wanted|use [[Definition:Value of Continued Fraction]]}} | Value of Finite Continued Fraction of Strictly Positive Real Numbers is Strictly Positive | https://proofwiki.org/wiki/Value_of_Finite_Continued_Fraction_of_Strictly_Positive_Real_Numbers_is_Strictly_Positive | https://proofwiki.org/wiki/Value_of_Finite_Continued_Fraction_of_Strictly_Positive_Real_Numbers_is_Strictly_Positive | [
"Continued Fractions"
] | [
"Definition:Continued Fraction/Finite",
"Definition:Length of Continued Fraction",
"Definition:Partial Denominator",
"Definition:Strictly Positive/Real Number",
"Definition:value of Continued Fraction"
] | [
"Definition:Value of Continued Fraction"
] |
proofwiki-13931 | Value of Finite Continued Fraction of Real Numbers is at Least First Term | Let $\sequence {a_0, \ldots, a_n}$ be a finite continued fraction in $\R$ of length $n \ge 0$.
Let the partial denominators $a_k > 0$ be strictly positive for $k>0$.
Let $x = [a_0, a_1, \ldots, a_n]$ be its value.
Then $x \ge a_0$, and $x > a_0$ if the length $n \ge 1$. | If $n = 0$, we have $x = \sqbrk {a_0} = a_0$ by definition of value.
Let $n>0$.
By definition of value:
:$\sqbrk {a_0, a_1, \ldots, a_n} = a_0 + \dfrac 1 {\sqbrk {a_1, a_2, \ldots, a_n} }$
By Value of Finite Continued Fraction of Strictly Positive Real Numbers is Strictly Positive:
:$\sqbrk {a_1, a_2, \ldots, a_n} > 0$... | Let $\sequence {a_0, \ldots, a_n}$ be a [[Definition:Finite Continued Fraction|finite continued fraction]] in $\R$ of [[Definition:Length of Continued Fraction|length]] $n \ge 0$.
Let the [[Definition:Partial Denominator|partial denominators]] $a_k > 0$ be [[Definition:Strictly Positive Real Number|strictly positive]]... | If $n = 0$, we have $x = \sqbrk {a_0} = a_0$ by definition of [[Definition:Value of Finite Continued Fraction|value]].
Let $n>0$.
By definition of [[Definition:Value of Finite Continued Fraction|value]]:
:$\sqbrk {a_0, a_1, \ldots, a_n} = a_0 + \dfrac 1 {\sqbrk {a_1, a_2, \ldots, a_n} }$
By [[Value of Finite Continu... | Value of Finite Continued Fraction of Real Numbers is at Least First Term | https://proofwiki.org/wiki/Value_of_Finite_Continued_Fraction_of_Real_Numbers_is_at_Least_First_Term | https://proofwiki.org/wiki/Value_of_Finite_Continued_Fraction_of_Real_Numbers_is_at_Least_First_Term | [
"Continued Fractions"
] | [
"Definition:Continued Fraction/Finite",
"Definition:Length of Continued Fraction",
"Definition:Partial Denominator",
"Definition:Strictly Positive/Real Number",
"Definition:Value of Continued Fraction",
"Definition:Length of Continued Fraction"
] | [
"Definition:Value of Continued Fraction/Finite",
"Definition:Value of Continued Fraction/Finite",
"Value of Finite Continued Fraction of Strictly Positive Real Numbers is Strictly Positive"
] |
proofwiki-13932 | Simple Finite Continued Fraction has Rational Value | Let $n \ge 0$ be a natural number.
Let $\tuple {a_0, \ldots, a_n}$ be a simple finite continued fraction of length $n$.
Then its value $\sqbrk {a_0, \ldots, a_n}$ is a rational number. | This will be proved by induction on the number of partial denominators.
For all $n \in \N$, let $\map P n$ be the proposition that the continued fraction $\sqbrk {a_0, a_1, \ldots, a_n}$ has a rational value. | Let $n \ge 0$ be a [[Definition:Natural Number|natural number]].
Let $\tuple {a_0, \ldots, a_n}$ be a [[Definition:Simple Finite Continued Fraction|simple finite continued fraction]] of [[Definition:Length of Continued Fraction|length]] $n$.
Then its [[Definition:Value of Finite Continued Fraction|value]] $\sqbrk {a... | This will be proved by [[Principle of Mathematical Induction|induction]] on the number of [[Definition:Partial Denominator|partial denominators]].
For all $n \in \N$, let $\map P n$ be the [[Definition:Proposition|proposition]] that the [[Definition:Continued Fraction|continued fraction]] $\sqbrk {a_0, a_1, \ldots, a... | Simple Finite Continued Fraction has Rational Value | https://proofwiki.org/wiki/Simple_Finite_Continued_Fraction_has_Rational_Value | https://proofwiki.org/wiki/Simple_Finite_Continued_Fraction_has_Rational_Value | [
"Simple Continued Fractions",
"Rational Numbers"
] | [
"Definition:Natural Numbers",
"Definition:Simple Continued Fraction/Finite",
"Definition:Length of Continued Fraction",
"Definition:Value of Continued Fraction/Finite",
"Definition:Rational Number"
] | [
"Principle of Mathematical Induction",
"Definition:Partial Denominator",
"Definition:Proposition",
"Definition:Continued Fraction",
"Definition:Rational Number",
"Definition:Value of Continued Fraction",
"Definition:Rational Number",
"Definition:Continued Fraction",
"Definition:Rational Number",
"... |
proofwiki-13933 | Accuracy of Convergents of Convergent Simple Infinite Continued Fraction | Let $C = \tuple {a_0, a_1, \ldots}$ be an simple infinite continued fraction in $\R$.
Let $C$ converge to $x \in \R$.
For $n \ge 0$, let $C_n = \dfrac {p_n} {q_n}$ be the $n$th convergent of $C$, where $p_n$ and $q_n$ are the $n$th numerator and denominator.
Then for all $n \ge 0$:
:$\size {x - \dfrac {p_n} {q_n} } < \... | We show that either:
:$x \in \closedint {C_n} {C_{n + 1} }$
or:
:$x \in \closedint {C_{n + 1} } {C_n}$
so that the result follows from:
:Difference between Adjacent Convergents of Simple Continued Fraction
:Distance between Point of Real Interval and Endpoint is at most Length | Let $C = \tuple {a_0, a_1, \ldots}$ be an [[Definition:Simple Infinite Continued Fraction|simple infinite continued fraction]] in $\R$.
Let $C$ [[Definition:Convergent Continued Fraction|converge]] to $x \in \R$.
For $n \ge 0$, let $C_n = \dfrac {p_n} {q_n}$ be the $n$th [[Definition:Convergent of Continued Fraction|... | We show that either:
:$x \in \closedint {C_n} {C_{n + 1} }$
or:
:$x \in \closedint {C_{n + 1} } {C_n}$
so that the result follows from:
:[[Difference between Adjacent Convergents of Simple Continued Fraction]]
:[[Distance between Point of Real Interval and Endpoint is at most Length]] | Accuracy of Convergents of Convergent Simple Infinite Continued Fraction | https://proofwiki.org/wiki/Accuracy_of_Convergents_of_Convergent_Simple_Infinite_Continued_Fraction | https://proofwiki.org/wiki/Accuracy_of_Convergents_of_Convergent_Simple_Infinite_Continued_Fraction | [
"Simple Continued Fractions"
] | [
"Definition:Simple Continued Fraction/Infinite",
"Definition:Convergent Continued Fraction",
"Definition:Convergent of Continued Fraction",
"Definition:Numerators and Denominators of Continued Fraction"
] | [
"Difference between Adjacent Convergents of Simple Continued Fraction",
"Distance between Point of Real Interval and Endpoint is at most Length"
] |
proofwiki-13934 | Locally Path-Connected Space is not necessarily Locally Injectively Path-Connected | Let $T = \struct {S, \tau}$ be a topological space which is locally path-connected.
Then it is not necessarily the case that $T$ is also a locally injectively path-connected space. | Let $T$ be the Either-Or topological space.
From Either-Or Topology is Locally Path-Connected, $T$ is a locally path-connected space.
From Either-Or Topology is not Locally Injectively Path-Connected, $T$ is not a locally injectively path-connected space.
Hence the result.
{{qed}} | Let $T = \struct {S, \tau}$ be a [[Definition:Topological Space|topological space]] which is [[Definition:Locally Path-Connected Space|locally path-connected]].
Then it is not necessarily the case that $T$ is also a [[Definition:Locally Injectively Path-Connected Space|locally injectively path-connected space]]. | Let $T$ be the [[Definition:Either-Or Topology|Either-Or topological space]].
From [[Either-Or Topology is Locally Path-Connected]], $T$ is a [[Definition:Locally Path-Connected Space|locally path-connected space]].
From [[Either-Or Topology is not Locally Injectively Path-Connected]], $T$ is not a [[Definition:Local... | Locally Path-Connected Space is not necessarily Locally Injectively Path-Connected | https://proofwiki.org/wiki/Locally_Path-Connected_Space_is_not_necessarily_Locally_Injectively_Path-Connected | https://proofwiki.org/wiki/Locally_Path-Connected_Space_is_not_necessarily_Locally_Injectively_Path-Connected | [
"Locally Path-Connected Spaces",
"Locally Injectively Path-Connected Spaces"
] | [
"Definition:Topological Space",
"Definition:Locally Path-Connected Space",
"Definition:Locally Injectively Path-Connected Space"
] | [
"Definition:Either-Or Topology",
"Either-Or Topology is Locally Path-Connected",
"Definition:Locally Path-Connected Space",
"Either-Or Topology is not Locally Injectively Path-Connected",
"Definition:Locally Injectively Path-Connected Space"
] |
proofwiki-13935 | Locally Connected Space is not necessarily Locally Path-Connected | Let $T = \struct {S, \tau}$ be a topological space which is locally connected.
Then it is not necessarily the case that $T$ is also an locally path-connected space. | Let $T$ be a countable finite complement Space.
From Finite Complement Space is Locally Connected, $T$ is a locally connected space.
From Countable Finite Complement Space is not Locally Path-Connected, $T$ is not a locally path-connected space.
Hence the result.
{{qed}} | Let $T = \struct {S, \tau}$ be a [[Definition:Topological Space|topological space]] which is [[Definition:Locally Connected Space|locally connected]].
Then it is not necessarily the case that $T$ is also an [[Definition:Locally Path-Connected Space|locally path-connected space]]. | Let $T$ be a [[Definition:Countable Finite Complement Topology|countable finite complement Space]].
From [[Finite Complement Space is Locally Connected]], $T$ is a [[Definition:Locally Connected Space|locally connected space]].
From [[Countable Finite Complement Space is not Locally Path-Connected]], $T$ is not a [[D... | Locally Connected Space is not necessarily Locally Path-Connected | https://proofwiki.org/wiki/Locally_Connected_Space_is_not_necessarily_Locally_Path-Connected | https://proofwiki.org/wiki/Locally_Connected_Space_is_not_necessarily_Locally_Path-Connected | [
"Locally Connected Spaces",
"Locally Path-Connected Spaces"
] | [
"Definition:Topological Space",
"Definition:Locally Connected Space",
"Definition:Locally Path-Connected Space"
] | [
"Definition:Finite Complement Topology/Countable",
"Finite Complement Space is Locally Connected",
"Definition:Locally Connected Space",
"Countable Finite Complement Space is not Locally Path-Connected",
"Definition:Locally Path-Connected Space"
] |
proofwiki-13936 | Path-Connected Space is not necessarily Locally Path-Connected | Let $T = \struct {S, \tau}$ be a topological space which is path-connected.
Then it is not necessarily the case that $T$ is also locally path-connected. | Let $T$ be the extended topologist's sine curve.
From Extended Topologist's Sine Curve is Path-Connected, $T$ is a path-connected space.
From Extended Topologist's Sine Curve is not Locally Path-Connected, $T$ is not a locally path-connected space.
Hence the result.
{{qed}} | Let $T = \struct {S, \tau}$ be a [[Definition:Topological Space|topological space]] which is [[Definition:Path-Connected Space|path-connected]].
Then it is not necessarily the case that $T$ is also [[Definition:Locally Path-Connected Space|locally path-connected]]. | Let $T$ be the [[Definition:Extended Topologist's Sine Curve|extended topologist's sine curve]].
From [[Extended Topologist's Sine Curve is Path-Connected]], $T$ is a [[Definition:Path-Connected Space|path-connected space]].
From [[Extended Topologist's Sine Curve is not Locally Path-Connected]], $T$ is not a [[Defin... | Path-Connected Space is not necessarily Locally Path-Connected | https://proofwiki.org/wiki/Path-Connected_Space_is_not_necessarily_Locally_Path-Connected | https://proofwiki.org/wiki/Path-Connected_Space_is_not_necessarily_Locally_Path-Connected | [
"Locally Path-Connected Spaces",
"Path-Connected Spaces"
] | [
"Definition:Topological Space",
"Definition:Path-Connected/Topological Space",
"Definition:Locally Path-Connected Space"
] | [
"Definition:Extended Topologist's Sine Curve",
"Extended Topologist's Sine Curve is Path-Connected",
"Definition:Path-Connected/Topological Space",
"Extended Topologist's Sine Curve is not Locally Path-Connected",
"Definition:Locally Path-Connected Space"
] |
proofwiki-13937 | Lower Bounds for Denominators of Simple Continued Fraction | Let $n \in \N \cup \{\infty\}$ be an extended natural number.
Let $\left[{a_0, a_1, a_2, \ldots}\right]$ be a simple continued fraction in $\R$ of length $N$.
Let $q_0, q_1, q_2, \ldots$ be its denominators. | By Denominators of Simple Continued Fraction are Strictly Increasing, with the possible exception of $1 = q_0 = q_1$, the sequence $\left \langle {q_n}\right \rangle$ is strictly increasing.
Now, since $q_2 > q_1 \ge q_0 = 1$, we have $q_2 \geq 2$.
Then $q_{k+1} \ge q_k + q_{k-1}$ shows that from $q_3$ onwards, the $q_... | Let $n \in \N \cup \{\infty\}$ be an [[Definition:Extended Natural Number|extended natural number]].
Let $\left[{a_0, a_1, a_2, \ldots}\right]$ be a [[Definition:Simple Continued Fraction|simple continued fraction]] in $\R$ of [[Definition:Length of Continued Fraction|length]] $N$.
Let $q_0, q_1, q_2, \ldots$ be its ... | By [[Denominators of Simple Continued Fraction are Strictly Increasing]], with the possible exception of $1 = q_0 = q_1$, the [[Definition:Sequence|sequence]] $\left \langle {q_n}\right \rangle$ is [[Definition:Strictly Increasing Sequence|strictly increasing]].
Now, since $q_2 > q_1 \ge q_0 = 1$, we have $q_2 \geq 2$... | Lower Bounds for Denominators of Simple Continued Fraction | https://proofwiki.org/wiki/Lower_Bounds_for_Denominators_of_Simple_Continued_Fraction | https://proofwiki.org/wiki/Lower_Bounds_for_Denominators_of_Simple_Continued_Fraction | [
"Simple Continued Fractions"
] | [
"Definition:Extended Natural Numbers",
"Definition:Simple Continued Fraction",
"Definition:Length of Continued Fraction",
"Definition:Numerators and Denominators of Continued Fraction"
] | [
"Denominators of Simple Continued Fraction are Strictly Increasing",
"Definition:Sequence",
"Definition:Strictly Increasing/Sequence",
"Category:Simple Continued Fractions"
] |
proofwiki-13938 | Equality of Rational Numbers | Let $a, b, c, d$ be integers, with $b$ and $d$ nonzero.
{{TFAE}}
:$(1): \quad$ The rational numbers $\dfrac a b$ and $\dfrac c d$ are equal.
:$(2): \quad$ The integers $a d$ and $b c$ are equal. | Note that by definition, $\Q$ is the field of quotients of $\Z$. | Let $a, b, c, d$ be [[Definition:Integer|integers]], with $b$ and $d$ [[Definition:Nonzero Integer|nonzero]].
{{TFAE}}
:$(1): \quad$ The [[Definition:Rational Number|rational numbers]] $\dfrac a b$ and $\dfrac c d$ are [[Definition:Equal|equal]].
:$(2): \quad$ The [[Definition:Integer|integers]] $a d$ and $b c$ are [... | Note that by definition, $\Q$ is the [[Definition:Field of Quotients|field of quotients]] of $\Z$. | Equality of Rational Numbers | https://proofwiki.org/wiki/Equality_of_Rational_Numbers | https://proofwiki.org/wiki/Equality_of_Rational_Numbers | [
"Rational Numbers",
"Equality"
] | [
"Definition:Integer",
"Definition:Nonzero Integer",
"Definition:Rational Number",
"Definition:Equals",
"Definition:Integer",
"Definition:Equals"
] | [
"Definition:Field of Quotients"
] |
proofwiki-13939 | Locally Path-Connected Space is not necessarily Path-Connected | Let $T = \struct {S, \tau}$ be a topological space which is locally path-connected.
Then it is not necessarily the case that $T$ is also path-connected. | Let $\struct {\R, \tau_d}$ be the real number line $\R$ under the usual (Euclidean) topology $\tau_d$.
Let $a, b, c \in \R$ where $a < b < c$.
Let $S$ be the union of the adjacent open intervals:
:$S := \openint a b \cup \openint b c$
Let $T := \struct {S, \tau_S}$ be the subspace composed of $S$ with the subspace topo... | Let $T = \struct {S, \tau}$ be a [[Definition:Topological Space|topological space]] which is [[Definition:Locally Path-Connected Space|locally path-connected]].
Then it is not necessarily the case that $T$ is also [[Definition:Path-Connected Space|path-connected]]. | Let $\struct {\R, \tau_d}$ be the [[Definition:Real Number Line|real number line]] $\R$ under the [[Definition:Euclidean Topology on Real Number Line|usual (Euclidean) topology]] $\tau_d$.
Let $a, b, c \in \R$ where $a < b < c$.
Let $S$ be the [[Definition:Union of Adjacent Open Intervals|union of the adjacent open i... | Locally Path-Connected Space is not necessarily Path-Connected | https://proofwiki.org/wiki/Locally_Path-Connected_Space_is_not_necessarily_Path-Connected | https://proofwiki.org/wiki/Locally_Path-Connected_Space_is_not_necessarily_Path-Connected | [
"Locally Path-Connected Spaces",
"Path-Connected Spaces"
] | [
"Definition:Topological Space",
"Definition:Locally Path-Connected Space",
"Definition:Path-Connected/Topological Space"
] | [
"Definition:Real Number/Real Number Line",
"Definition:Euclidean Space/Euclidean Topology/Real Number Line",
"Definition:Union of Adjacent Open Intervals",
"Definition:Topological Subspace",
"Definition:Topological Subspace",
"Union of Adjacent Open Intervals is Locally Path-Connected",
"Definition:Loca... |
proofwiki-13940 | Injectively Path-Connected Space is not necessarily Locally Injectively Path-Connected | Let $T = \struct {S, \tau}$ be a topological space which is injectively path-connected.
Then it is not necessarily the case that $T$ is also locally arc-connected. | Let $T$ be the extended topologist's sine curve.
From Extended Topologist's Sine Curve is Injectively Path-Connected, $T$ is an injectively path-connected space.
From Extended Topologist's Sine Curve is not Locally Injectively Path-Connected, $T$ is not a locally injectively path-connected space.
Hence the result.
{{qe... | Let $T = \struct {S, \tau}$ be a [[Definition:Topological Space|topological space]] which is [[Definition:Injectively Path-Connected Space|injectively path-connected]].
Then it is not necessarily the case that $T$ is also [[Definition:Locally Injectively Path-Connected Space|locally arc-connected]]. | Let $T$ be the [[Definition:Extended Topologist's Sine Curve|extended topologist's sine curve]].
From [[Extended Topologist's Sine Curve is Injectively Path-Connected]], $T$ is an [[Definition:Injectively Path-Connected Space|injectively path-connected space]].
From [[Extended Topologist's Sine Curve is not Locally I... | Injectively Path-Connected Space is not necessarily Locally Injectively Path-Connected | https://proofwiki.org/wiki/Injectively_Path-Connected_Space_is_not_necessarily_Locally_Injectively_Path-Connected | https://proofwiki.org/wiki/Injectively_Path-Connected_Space_is_not_necessarily_Locally_Injectively_Path-Connected | [
"Locally Injectively Path-Connected Spaces",
"Injectively Path-Connected Spaces"
] | [
"Definition:Topological Space",
"Definition:Injectively Path-Connected/Topological Space",
"Definition:Locally Injectively Path-Connected Space"
] | [
"Definition:Extended Topologist's Sine Curve",
"Extended Topologist's Sine Curve is Injectively Path-Connected",
"Definition:Injectively Path-Connected/Topological Space",
"Extended Topologist's Sine Curve is not Locally Injectively Path-Connected",
"Definition:Locally Injectively Path-Connected Space"
] |
proofwiki-13941 | Locally Injectively Path-Connected Space is not necessarily Injectively Path-Connected | Let $T = \struct {S, \tau}$ be a topological space which is locally injectively path-connected.
Then it is not necessarily the case that $T$ is also injectively path-connected. | Let $\struct {\R, \tau_d}$ be the real number line $\R$ under the usual (Euclidean) topology $\tau_d$.
Let $a, b, c \in \R$ where $a < b < c$.
Let $S$ be the union of the adjacent open intervals:
:$S := \openint a b \cup \openint b c$
Let $T := \struct {S, \tau_S}$ be the subspace composed of $S$ with the subspace topo... | Let $T = \struct {S, \tau}$ be a [[Definition:Topological Space|topological space]] which is [[Definition:Locally Injectively Path-Connected Space|locally injectively path-connected]].
Then it is not necessarily the case that $T$ is also [[Definition:Injectively Path-Connected Space|injectively path-connected]]. | Let $\struct {\R, \tau_d}$ be the [[Definition:Real Number Line|real number line]] $\R$ under the [[Definition:Euclidean Topology on Real Number Line|usual (Euclidean) topology]] $\tau_d$.
Let $a, b, c \in \R$ where $a < b < c$.
Let $S$ be the [[Definition:Union of Adjacent Open Intervals|union of the adjacent open i... | Locally Injectively Path-Connected Space is not necessarily Injectively Path-Connected | https://proofwiki.org/wiki/Locally_Injectively_Path-Connected_Space_is_not_necessarily_Injectively_Path-Connected | https://proofwiki.org/wiki/Locally_Injectively_Path-Connected_Space_is_not_necessarily_Injectively_Path-Connected | [
"Locally Injectively Path-Connected Spaces",
"Injectively Path-Connected Spaces"
] | [
"Definition:Topological Space",
"Definition:Locally Injectively Path-Connected Space",
"Definition:Injectively Path-Connected/Topological Space"
] | [
"Definition:Real Number/Real Number Line",
"Definition:Euclidean Space/Euclidean Topology/Real Number Line",
"Definition:Union of Adjacent Open Intervals",
"Definition:Topological Subspace",
"Definition:Topological Subspace",
"Union of Adjacent Open Intervals is Locally Injectively Path-Connected",
"Def... |
proofwiki-13942 | Odd Convergents of Simple Continued Fraction are Strictly Decreasing | The odd convergents satisfy $C_1 > C_3 > C_5 > \cdots$ | Let $ k \ge 3$ be an odd integer.
From Difference between Adjacent Convergents But One of Simple Continued Fraction:
:$C_k - C_{k - 2} = \dfrac {\paren {-1}^k a_k} {q_k q_{k - 2} } = \dfrac {-a_k} {q_k q_{k - 2} }$
By definition of simple continued fraction, $a_k > 0$.
By Convergents of Simple Continued Fraction are Ra... | The [[Definition:Odd Convergent|odd convergents]] satisfy $C_1 > C_3 > C_5 > \cdots$ | Let $ k \ge 3$ be an [[Definition:Odd Integer|odd integer]].
From [[Difference between Adjacent Convergents But One of Simple Continued Fraction]]:
:$C_k - C_{k - 2} = \dfrac {\paren {-1}^k a_k} {q_k q_{k - 2} } = \dfrac {-a_k} {q_k q_{k - 2} }$
By definition of [[Definition:Simple Continued Fraction|simple continued... | Odd Convergents of Simple Continued Fraction are Strictly Decreasing | https://proofwiki.org/wiki/Odd_Convergents_of_Simple_Continued_Fraction_are_Strictly_Decreasing | https://proofwiki.org/wiki/Odd_Convergents_of_Simple_Continued_Fraction_are_Strictly_Decreasing | [
"Simple Continued Fractions"
] | [
"Definition:Convergent of Continued Fraction/Odd"
] | [
"Definition:Odd Integer",
"Difference between Adjacent Convergents But One of Simple Continued Fraction",
"Definition:Simple Continued Fraction",
"Convergents of Simple Continued Fraction are Rationals in Canonical Form"
] |
proofwiki-13943 | Even Convergents of Simple Continued Fraction are Strictly Increasing | The even convergents satisfy $C_0 < C_2 < C_4 \cdots$. | Let $k \ge 2$ be an even integer.
From Difference between Adjacent Convergents But One of Simple Continued Fraction:
:$C_k - C_{k - 2} = \dfrac {\paren {-1}^k a_k} {q_k q_{k - 2} } = \dfrac {a_k} {q_k q_{k - 2} }$
By definition of simple continued fraction, $a_k > 0$.
By Convergents of Simple Continued Fraction are Rat... | The [[Definition:Even Convergent|even convergents]] satisfy $C_0 < C_2 < C_4 \cdots$. | Let $k \ge 2$ be an [[Definition:Even Integer|even integer]].
From [[Difference between Adjacent Convergents But One of Simple Continued Fraction]]:
:$C_k - C_{k - 2} = \dfrac {\paren {-1}^k a_k} {q_k q_{k - 2} } = \dfrac {a_k} {q_k q_{k - 2} }$
By definition of [[Definition:Simple Continued Fraction|simple continued... | Even Convergents of Simple Continued Fraction are Strictly Increasing | https://proofwiki.org/wiki/Even_Convergents_of_Simple_Continued_Fraction_are_Strictly_Increasing | https://proofwiki.org/wiki/Even_Convergents_of_Simple_Continued_Fraction_are_Strictly_Increasing | [
"Simple Continued Fractions"
] | [
"Definition:Convergent of Continued Fraction/Even"
] | [
"Definition:Even Integer",
"Difference between Adjacent Convergents But One of Simple Continued Fraction",
"Definition:Simple Continued Fraction",
"Convergents of Simple Continued Fraction are Rationals in Canonical Form"
] |
proofwiki-13944 | Denominators of Simple Continued Fraction are Strictly Positive | Let $n \in \N \cup \set \infty$ be an extended natural number.
Let $\tuple {a_0, a_1, \ldots}$ be a simple continued fraction in $\R$ of length $n$.
Let $q_0, q_1, q_2, \ldots$ be its denominators.
Then for $0 \leq k \leq n$ we have $q_k > 0$. | Follows from:
:$q_0 = 1$ by definition
:Denominators of Simple Continued Fraction are Strictly Increasing.
{{qed}}
Category:Simple Continued Fractions
rwuh6ocjib4wj0e04jrij048mh593gw | Let $n \in \N \cup \set \infty$ be an [[Definition:Extended Natural Number|extended natural number]].
Let $\tuple {a_0, a_1, \ldots}$ be a [[Definition:Simple Continued Fraction|simple continued fraction]] in $\R$ of [[Definition:Length of Continued Fraction|length]] $n$.
Let $q_0, q_1, q_2, \ldots$ be its [[Definiti... | Follows from:
:$q_0 = 1$ by definition
:[[Denominators of Simple Continued Fraction are Strictly Increasing]].
{{qed}}
[[Category:Simple Continued Fractions]]
rwuh6ocjib4wj0e04jrij048mh593gw | Denominators of Simple Continued Fraction are Strictly Positive | https://proofwiki.org/wiki/Denominators_of_Simple_Continued_Fraction_are_Strictly_Positive | https://proofwiki.org/wiki/Denominators_of_Simple_Continued_Fraction_are_Strictly_Positive | [
"Simple Continued Fractions"
] | [
"Definition:Extended Natural Numbers",
"Definition:Simple Continued Fraction",
"Definition:Length of Continued Fraction",
"Definition:Numerators and Denominators of Continued Fraction"
] | [
"Denominators of Simple Continued Fraction are Strictly Increasing",
"Category:Simple Continued Fractions"
] |
proofwiki-13945 | Simple Finite Continued Fraction is Almost Determined by Value | Let $n,m \geq 0$ be natural number.
Let $\sequence {a_k}_{0 \mathop \le k \mathop \le m}$ and $\sequence {b_k}_{0 \mathop \le k \mathop \le n}$ be simple finite continued fractions in $\R$.
Let $\sequence {a_k}_{0 \mathop \le k \mathop \le m}$ and $\sequence {b_k}_{0 \mathop \le k \mathop \le n}$ have the same value.
T... | {{proof wanted|use Floor of Simple Finite Continued Fraction}} | Let $n,m \geq 0$ be [[Definition:Natural Number|natural number]].
Let $\sequence {a_k}_{0 \mathop \le k \mathop \le m}$ and $\sequence {b_k}_{0 \mathop \le k \mathop \le n}$ be [[Definition:Simple Finite Continued Fraction|simple finite continued fractions]] in $\R$.
Let $\sequence {a_k}_{0 \mathop \le k \mathop \le ... | {{proof wanted|use [[Floor of Simple Finite Continued Fraction]]}} | Simple Finite Continued Fraction is Almost Determined by Value | https://proofwiki.org/wiki/Simple_Finite_Continued_Fraction_is_Almost_Determined_by_Value | https://proofwiki.org/wiki/Simple_Finite_Continued_Fraction_is_Almost_Determined_by_Value | [
"Simple Continued Fractions"
] | [
"Definition:Natural Numbers",
"Definition:Simple Continued Fraction/Finite",
"Definition:Value of Continued Fraction/Finite",
"Definition:Finite Sequence"
] | [
"Floor of Simple Finite Continued Fraction"
] |
proofwiki-13946 | Subgroup Generated by One Element is Set of Powers | Let $G$ be a group.
Let $a \in G$.
Then the subgroup generated by $a$ is the set of powers:
:$\gen a = \set {a^n : n \in \Z}$ | By definition, the subgroup generated by $a$ is the intersection of all subgroups containing $a$.
By Powers of Element form Subgroup, the set $H = \set {a^n : n \in \Z}$ is a subgroup.
Thus $\gen a \subseteq H$.
By Power of Element in Subgroup, $H \subseteq \gen a$.
By definition of set equality, $\gen a = H$.
{{qed}} | Let $G$ be a [[Definition:group|group]].
Let $a \in G$.
Then the [[Definition:Generated Subgroup|subgroup generated]] by $a$ is the [[Definition:Set|set]] of [[Definition:Power of Group Element|powers]]:
:$\gen a = \set {a^n : n \in \Z}$ | By definition, the [[Definition:Generated Subgroup|subgroup generated]] by $a$ is the [[Definition:Set Intersection|intersection]] of all [[Definition:Subgroup|subgroups]] containing $a$.
By [[Powers of Element form Subgroup]], the set $H = \set {a^n : n \in \Z}$ is a [[Definition:Subgroup|subgroup]].
Thus $\gen a \s... | Subgroup Generated by One Element is Set of Powers | https://proofwiki.org/wiki/Subgroup_Generated_by_One_Element_is_Set_of_Powers | https://proofwiki.org/wiki/Subgroup_Generated_by_One_Element_is_Set_of_Powers | [
"Generated Subgroups"
] | [
"Definition:group",
"Definition:Generated Subgroup",
"Definition:Set",
"Definition:Power of Element/Group"
] | [
"Definition:Generated Subgroup",
"Definition:Set Intersection",
"Definition:Subgroup",
"Powers of Element form Subgroup",
"Definition:Subgroup",
"Power of Element in Subgroup",
"Definition:Set Equality"
] |
proofwiki-13947 | Finite Product Space is Connected iff Factors are Connected | Let $T_1 = \struct {S_1, \tau_1}, T_2 = \struct {S_2, \tau_2}, \dotsc, T_n = \struct {S_n, \tau_n}$ be topological spaces.
Let $T = \ds \prod_{i \mathop = 1}^n T_i$ be the product space of $T_1, T_2, \ldots, T_n$.
Then $T$ is connected {{iff}} each of $T_1, T_2, \ldots, T_n$ are connected. | The proof proceeds by induction.
For all $n \in \Z_{\ge 2}$, let $\map P n$ be the proposition:
:$T$ is connected {{iff}} each of $T_1, T_2, \ldots, T_n$ are connected. | Let $T_1 = \struct {S_1, \tau_1}, T_2 = \struct {S_2, \tau_2}, \dotsc, T_n = \struct {S_n, \tau_n}$ be [[Definition:Topological Space|topological spaces]].
Let $T = \ds \prod_{i \mathop = 1}^n T_i$ be the [[Definition:Product Space|product space]] of $T_1, T_2, \ldots, T_n$.
Then $T$ is [[Definition:Connected Topolo... | The proof proceeds by [[Principle of Mathematical Induction|induction]].
For all $n \in \Z_{\ge 2}$, let $\map P n$ be the [[Definition:Proposition|proposition]]:
:$T$ is [[Definition:Connected Topological Space|connected]] {{iff}} each of $T_1, T_2, \ldots, T_n$ are [[Definition:Connected Topological Space|connected]... | Finite Product Space is Connected iff Factors are Connected | https://proofwiki.org/wiki/Finite_Product_Space_is_Connected_iff_Factors_are_Connected | https://proofwiki.org/wiki/Finite_Product_Space_is_Connected_iff_Factors_are_Connected | [
"Product Space is Connected iff Factors are Connected",
"Product Spaces",
"Connected Topological Spaces"
] | [
"Definition:Topological Space",
"Definition:Product Space",
"Definition:Connected Topological Space",
"Definition:Connected Topological Space"
] | [
"Principle of Mathematical Induction",
"Definition:Proposition",
"Definition:Connected Topological Space",
"Definition:Connected Topological Space",
"Definition:Connected Topological Space",
"Definition:Connected Topological Space",
"Definition:Connected Topological Space",
"Definition:Connected Topol... |
proofwiki-13948 | Equivalence of Definitions of Limit of Vector-Valued Function | Let $D \subseteq \R$ be a subset and $f: D \to \R^n, \map f x = \tuple {\map {f_1} x, \ldots, \map {f_n} x}$ a vector valued function.
Let $x_0 \in \R$ be a limit point of $D$ and $L = (L_1,\ldots,L_n) \in \R^n$.
Then $\ds \lim_{x \mathop \to x_0} \map f x = L$ {{iff}} $\ds \lim_{x \mathop \to x_0} \map {f_j} x = L_j$
... | === Sufficient Condition ===
First assume that $\ds \lim_{x \mathop \to x_0} \map f x = L$.
Let $\epsilon \in \R_{\gt 0}$.
Then there exists $\delta \in \R_{\gt 0}$ such that for all $x \in D$ with $\size {x - x_0} \lt \delta$ we have $\size {\map f x - L} \lt \epsilon$.
Then it follows for all $j = \set {1, \ldots, n}... | Let $D \subseteq \R$ be a subset and $f: D \to \R^n, \map f x = \tuple {\map {f_1} x, \ldots, \map {f_n} x}$ a vector valued function.
Let $x_0 \in \R$ be a limit point of $D$ and $L = (L_1,\ldots,L_n) \in \R^n$.
Then $\ds \lim_{x \mathop \to x_0} \map f x = L$ {{iff}} $\ds \lim_{x \mathop \to x_0} \map {f_j} x = L_... | === Sufficient Condition ===
First assume that $\ds \lim_{x \mathop \to x_0} \map f x = L$.
Let $\epsilon \in \R_{\gt 0}$.
Then there exists $\delta \in \R_{\gt 0}$ such that for all $x \in D$ with $\size {x - x_0} \lt \delta$ we have $\size {\map f x - L} \lt \epsilon$.
Then it follows for all $j = \set {1, \ldots... | Equivalence of Definitions of Limit of Vector-Valued Function | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Limit_of_Vector-Valued_Function | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Limit_of_Vector-Valued_Function | [
"Limits of Mappings"
] | [] | [] |
proofwiki-13949 | Binary Product in Preadditive Category is Biproduct | Let $A$ be a preadditive category.
Let $a_1, a_2$ be objects of $A$.
Let $(a_1 \times a_2, p_1, p_2)$ be their binary product, assuming it exists.
Let $i_1 : a_1 \to a_1 \times a_2$ be the unique morphism with:
:$p_1 \circ i_1 = 1 : a_1 \to a_1$
:$p_2 \circ i_1 = 0 : a_1 \to a_2$
Let $i_2 : a_1 \to a_1 \times a_2$ be t... | By definition of binary biproduct, it remains to verify that:
:$i_1\circ p_1 + i_2 \circ p_2 = 1 : a_1 \times a_2 \to a_1 \times a_2$.
By definition of binary product and identity morphism, $1 : a_1 \times a_2 \to a_1 \times a_2$ is the unique morphism with:
:$p_1 \circ 1 = p_1$
:$p_2 \circ 1 = p_2$
Thus it remains to ... | Let $A$ be a [[Definition:Preadditive Category|preadditive category]].
Let $a_1, a_2$ be [[Definition:Object of Category|objects]] of $A$.
Let $(a_1 \times a_2, p_1, p_2)$ be their [[Definition:Binary Product (Category Theory)|binary product]], assuming it exists.
Let $i_1 : a_1 \to a_1 \times a_2$ be the [[Definiti... | By definition of [[Definition:Binary Biproduct|binary biproduct]], it remains to verify that:
:$i_1\circ p_1 + i_2 \circ p_2 = 1 : a_1 \times a_2 \to a_1 \times a_2$.
By definition of [[Definition:Binary Product (Category Theory)|binary product]] and [[Definition:Identity Morphism|identity morphism]], $1 : a_1 \times ... | Binary Product in Preadditive Category is Biproduct | https://proofwiki.org/wiki/Binary_Product_in_Preadditive_Category_is_Biproduct | https://proofwiki.org/wiki/Binary_Product_in_Preadditive_Category_is_Biproduct | [
"Preadditive Categories"
] | [
"Definition:Preadditive Category",
"Definition:Object (Category Theory)",
"Definition:Product (Category Theory)/Binary Product",
"Definition:Unique",
"Definition:Morphism",
"Definition:Unique",
"Definition:Morphism",
"Definition:Identity Morphism",
"Definition:Zero Morphism in Preadditive Category",... | [
"Definition:Binary Biproduct",
"Definition:Product (Category Theory)/Binary Product",
"Definition:Identity Morphism",
"Definition:Unique",
"Definition:Morphism"
] |
proofwiki-13950 | Binary Coproduct in Preadditive Category is Biproduct | Let $A$ be a preadditive category.
Let $a_1, a_2$ be objects of $A$.
Let $(a_1 \sqcup a_2, i_1, i_2)$ be their binary coproduct, assuming it exists.
Let $p_1 : a_1 \sqcup a_2 \to a_1$ be the unique morphism with:
:$p_1 \circ i_1 = 1 : a_1 \to a_1$
:$p_1 \circ i_2 = 0 : a_1 \to a_2$
Let $p_2 : a_1 \sqcup a_2 \to a_2$ be... | By definition of binary biproduct, it remains to verify that:
:$i_1\circ p_1 + i_2 \circ p_2 = 1 : a_1 \sqcup a_2 \to a_1 \sqcup a_2$.
By definition of binary coproduct and identity morphism, $1 : a_1 \sqcup a_2 \to a_1 \sqcup a_2$ is the unique morphism with:
:$1 \circ i_1 = i_1$
:$1 \circ i_2 = i_2$
Thus it remains t... | Let $A$ be a [[Definition:Preadditive Category|preadditive category]].
Let $a_1, a_2$ be [[Definition:Object of Category|objects]] of $A$.
Let $(a_1 \sqcup a_2, i_1, i_2)$ be their [[Definition:Binary Coproduct (Category Theory)|binary coproduct]], assuming it exists.
Let $p_1 : a_1 \sqcup a_2 \to a_1$ be the [[Defi... | By definition of [[Definition:Binary Biproduct|binary biproduct]], it remains to verify that:
:$i_1\circ p_1 + i_2 \circ p_2 = 1 : a_1 \sqcup a_2 \to a_1 \sqcup a_2$.
By definition of [[Definition:Binary Coproduct (Category Theory)|binary coproduct]] and [[Definition:Identity Morphism|identity morphism]], $1 : a_1 \sq... | Binary Coproduct in Preadditive Category is Biproduct | https://proofwiki.org/wiki/Binary_Coproduct_in_Preadditive_Category_is_Biproduct | https://proofwiki.org/wiki/Binary_Coproduct_in_Preadditive_Category_is_Biproduct | [
"Coproducts",
"Preadditive Categories"
] | [
"Definition:Preadditive Category",
"Definition:Object (Category Theory)",
"Definition:Coproduct",
"Definition:Unique",
"Definition:Morphism",
"Definition:Unique",
"Definition:Morphism",
"Definition:Identity Morphism",
"Definition:Zero Morphism in Preadditive Category",
"Definition:Binary Biproduct... | [
"Definition:Binary Biproduct",
"Definition:Coproduct",
"Definition:Identity Morphism",
"Definition:Unique",
"Definition:Morphism"
] |
proofwiki-13951 | Product Space is Connected iff Factors are Connected | Let $I$ be an indexing set.
Let $\family {T_\alpha}_{\alpha \mathop \in I}$ be an indexed family of topological spaces.
Let $T = \ds \prod_{\alpha \mathop \in I} T_\alpha$ be the Cartesian space of $\family {T_\alpha}_{\alpha \mathop \in I}$.
Let:
:$T = \ds \overline {\bigcup_{\alpha \mathop \in I} S_\alpha}$
where $\d... | Let the {{Axiom-link|Choice}} be assumed.
Let $I$ be well-ordered.
Let $x = \family {x_\alpha} \in T$ be some arbitrary fixed element of $T$.
Let $S_\alpha = \set {\family {y_\beta} \in T: y_\beta = x_\beta \text { for all } \beta \ge \alpha}$.
We have that $S_\alpha$ is homeomorphic to $S_{\alpha - 1} \times T_\alpha$... | Let $I$ be an [[Definition:Indexing Set|indexing set]].
Let $\family {T_\alpha}_{\alpha \mathop \in I}$ be an [[Definition:Indexed Family|indexed family]] of [[Definition:Topological Space|topological spaces]].
Let $T = \ds \prod_{\alpha \mathop \in I} T_\alpha$ be the [[Definition:Cartesian Space|Cartesian space]] o... | Let the {{Axiom-link|Choice}} be assumed.
Let $I$ be [[Definition:Well-Ordered Set|well-ordered]].
Let $x = \family {x_\alpha} \in T$ be some arbitrary fixed [[Definition:Element|element]] of $T$.
Let $S_\alpha = \set {\family {y_\beta} \in T: y_\beta = x_\beta \text { for all } \beta \ge \alpha}$.
We have that $S_... | Product Space is Connected iff Factors are Connected | https://proofwiki.org/wiki/Product_Space_is_Connected_iff_Factors_are_Connected | https://proofwiki.org/wiki/Product_Space_is_Connected_iff_Factors_are_Connected | [
"Product Space is Connected iff Factors are Connected",
"Product Spaces",
"Connected Topological Spaces"
] | [
"Definition:Indexing Set",
"Definition:Indexing Set/Family",
"Definition:Topological Space",
"Definition:Cartesian Product/Cartesian Space",
"Definition:Closure (Topology)",
"Definition:Topology",
"Definition:Homeomorphism/Topological Spaces",
"Definition:Connected Topological Space",
"Definition:Co... | [
"Definition:Well-Ordered Set",
"Definition:Element",
"Definition:Homeomorphism/Topological Spaces",
"Finite Product Space is Connected iff Factors are Connected",
"Definition:Connected Set (Topology)",
"Definition:Limit Ordinal",
"Definition:Closure (Topology)",
"Definition:Connected Set (Topology)",
... |
proofwiki-13952 | Continuous Mapping from Compact Space to Hausdorff Space Preserves Local Connectedness | Let $T_1 = \struct {S_1, \tau_1}$ be a compact topological space.
Let $T_2 = \struct {S_2, \tau_2}$ be a $T_2$ (Hausdorff) space.
Let $f: T_1 \to T_2$ be a continuous mapping.
Let $T_1$ be locally connected.
Then $T_2$ is also locally connected. | Let $H$ be a component of an open set $U$ of $T_2$.
By definition of continuous mapping, $f^{-1} \sqbrk U$ is an open set of $T_1$.
Let $G$ be a component of $f^{-1} \sqbrk U$.
Thus by Continuous Image of Connected Space is Connected, $f \sqbrk G$ is connected in $T_2$.
Thus either:
:$f \sqbrk G \subseteq H$
or:
:$f \s... | Let $T_1 = \struct {S_1, \tau_1}$ be a [[Definition:Compact Topological Space|compact topological space]].
Let $T_2 = \struct {S_2, \tau_2}$ be a [[Definition:Hausdorff Space|$T_2$ (Hausdorff) space]].
Let $f: T_1 \to T_2$ be a [[Definition:Everywhere Continuous Mapping (Topology)|continuous mapping]].
Let $T_1$ be ... | Let $H$ be a [[Definition:Component (Topology)|component]] of an [[Definition:Open Set (Topology)|open set]] $U$ of $T_2$.
By definition of [[Definition:Everywhere Continuous Mapping (Topology)|continuous mapping]], $f^{-1} \sqbrk U$ is an [[Definition:Open Set (Topology)|open set]] of $T_1$.
Let $G$ be a [[Definitio... | Continuous Mapping from Compact Space to Hausdorff Space Preserves Local Connectedness | https://proofwiki.org/wiki/Continuous_Mapping_from_Compact_Space_to_Hausdorff_Space_Preserves_Local_Connectedness | https://proofwiki.org/wiki/Continuous_Mapping_from_Compact_Space_to_Hausdorff_Space_Preserves_Local_Connectedness | [
"Locally Connected Spaces",
"Compact Topological Spaces",
"Hausdorff Spaces",
"Continuous Mappings"
] | [
"Definition:Compact Topological Space",
"Definition:T2 Space",
"Definition:Continuous Mapping (Topology)/Everywhere",
"Definition:Locally Connected Space",
"Definition:Locally Connected Space"
] | [
"Definition:Component (Topology)",
"Definition:Open Set/Topology",
"Definition:Continuous Mapping (Topology)/Everywhere",
"Definition:Open Set/Topology",
"Definition:Component (Topology)",
"Continuous Image of Connected Space is Connected",
"Definition:Connected Set (Topology)",
"Definition:Component ... |
proofwiki-13953 | Minimum Degree Bound for Simple Planar Graph | Let $G$ be a simple connected planar graph.
Then:
:$\map \delta G \le 5$
where $\map \delta G$ denotes the minimum degree of vertices of $G$. | {{AimForCont}} $G$ is a simple connected planar graph with $\map \delta G \ge 6$.
Let $m$ and $n$ denote the number of edges and vertices respectively in $G$.
Then by the Handshake Lemma:
{{begin-eqn}}
{{eqn | l = 2 m
| r = \sum_{i \mathop = 1}^n \map {\deg_G} {v_i}
| c = where $\map {\deg_G} {v_i}$ is t... | Let $G$ be a [[Definition:Simple Graph|simple]] [[Definition:Connected Graph|connected]] [[Definition:Planar Graph|planar graph]].
Then:
:$\map \delta G \le 5$
where $\map \delta G$ denotes the minimum [[Definition:Degree of Vertex|degree of vertices]] of $G$. | {{AimForCont}} $G$ is a [[Definition:Simple Graph|simple]] [[Definition:Connected Graph|connected]] [[Definition:Planar Graph|planar graph]] with $\map \delta G \ge 6$.
Let $m$ and $n$ denote the number of [[Definition:Edge of Graph|edges]] and [[Definition:Vertex of Graph|vertices]] respectively in $G$.
Then by t... | Minimum Degree Bound for Simple Planar Graph | https://proofwiki.org/wiki/Minimum_Degree_Bound_for_Simple_Planar_Graph | https://proofwiki.org/wiki/Minimum_Degree_Bound_for_Simple_Planar_Graph | [
"Graph Theory"
] | [
"Definition:Simple Graph",
"Definition:Connected (Graph Theory)/Graph",
"Definition:Planar Graph",
"Definition:Degree of Vertex"
] | [
"Definition:Simple Graph",
"Definition:Connected (Graph Theory)/Graph",
"Definition:Planar Graph",
"Definition:Graph (Graph Theory)/Edge",
"Definition:Graph (Graph Theory)/Vertex",
"Handshake Lemma",
"Definition:Degree of Vertex",
"Definition:Graph (Graph Theory)/Vertex",
"Definition:Contradiction",... |
proofwiki-13954 | Linear Bound Lemma | Let $G_n$ be a simple connected planar graph with $n$ vertices.
Then:
:$m \le 3 n − 6$
where $m$ is the number of edges. | Let $f$ denote the number of faces of $G_n$.
Let $\sequence {s_i}_{i \mathop = 1}^f$ be a sequence of regions of a planar embedding of $G_n$.
Consider the sequence $\sequence {r_i}_{i \mathop = 1}^f$ where $r_i$ denotes the number of boundary edges of $s_i$.
Since $G$ is simple, then by the definition of planar embe... | Let $G_n$ be a [[Definition:Simple Graph|simple]] [[Definition:Connected Graph|connected]] [[Definition:Planar Graph|planar graph]] with $n$ [[Definition:Vertex of Graph|vertices]].
Then:
:$m \le 3 n − 6$
where $m$ is the number of [[Definition:Edge of Graph|edges]]. | Let $f$ denote the number of [[Definition:Face of Graph|faces]] of $G_n$.
Let $\sequence {s_i}_{i \mathop = 1}^f$ be a [[Definition:Sequence|sequence]] of [[Definition:Region of Planar Embedding|regions]] of a [[Definition:Planar Embedding|planar embedding]] of $G_n$.
Consider the [[Definition:Sequence|sequence]] $... | Linear Bound Lemma | https://proofwiki.org/wiki/Linear_Bound_Lemma | https://proofwiki.org/wiki/Linear_Bound_Lemma | [
"Graph Theory",
"Named Theorems"
] | [
"Definition:Simple Graph",
"Definition:Connected (Graph Theory)/Graph",
"Definition:Planar Graph",
"Definition:Graph (Graph Theory)/Vertex",
"Definition:Graph (Graph Theory)/Edge"
] | [
"Definition:Planar Graph/Face",
"Definition:Sequence",
"Definition:Region of Planar Embedding",
"Definition:Planar Embedding",
"Definition:Sequence",
"Definition:Boundary Edge",
"Definition:Simple Graph",
"Definition:Planar Embedding",
"Definition:Region of Planar Embedding",
"Definition:Boundary ... |
proofwiki-13955 | Existence of Connected Space which is Totally Pathwise Disconnected | There exists at least one example of a topological space which is both connected and totally pathwise disconnected. | Let $T$ be Cantor's leaky tent.
From Cantor's Leaky Tent is Connected, $T$ is a connected space.
From Cantor's Leaky Tent is Totally Pathwise Disconnected, $T$ is a totally pathwise disconnected space.
Hence the result.
{{qed}} | There exists at least one example of a [[Definition:Topological Space|topological space]] which is both [[Definition:Connected Topological Space|connected]] and [[Definition:Totally Pathwise Disconnected Space|totally pathwise disconnected]]. | Let $T$ be [[Definition:Cantor's Leaky Tent|Cantor's leaky tent]].
From [[Cantor's Leaky Tent is Connected]], $T$ is a [[Definition:Connected Topological Space|connected space]].
From [[Cantor's Leaky Tent is Totally Pathwise Disconnected]], $T$ is a [[Definition:Totally Pathwise Disconnected Space|totally pathwise ... | Existence of Connected Space which is Totally Pathwise Disconnected | https://proofwiki.org/wiki/Existence_of_Connected_Space_which_is_Totally_Pathwise_Disconnected | https://proofwiki.org/wiki/Existence_of_Connected_Space_which_is_Totally_Pathwise_Disconnected | [
"Connected Topological Spaces",
"Totally Pathwise Disconnected Spaces"
] | [
"Definition:Topological Space",
"Definition:Connected Topological Space",
"Definition:Totally Pathwise Disconnected Space"
] | [
"Definition:Cantor's Leaky Tent",
"Cantor's Leaky Tent is Connected",
"Definition:Connected Topological Space",
"Cantor's Leaky Tent is Totally Pathwise Disconnected",
"Definition:Totally Pathwise Disconnected Space"
] |
proofwiki-13956 | Zero Dimensional Space is not necessarily T0 | Let $T = \struct {S, \tau}$ be a zero dimensional topological space.
Then $T$ is not necessarily a $T_0$ (Kolmogorov) space. | Let $T = \struct {S, \tau}$ be a partition space.
From Partition Topology is Zero Dimensional, $T$ is a zero dimensional topological space.
From Partition Topology is not $T_0$, $T$ is not a $T_0$ (Kolmogorov) space.
{{qed}} | Let $T = \struct {S, \tau}$ be a [[Definition:Zero Dimensional Space|zero dimensional topological space]].
Then $T$ is not necessarily a [[Definition:T0 Space|$T_0$ (Kolmogorov) space]]. | Let $T = \struct {S, \tau}$ be a [[Definition:Partition Space|partition space]].
From [[Partition Topology is Zero Dimensional]], $T$ is a [[Definition:Zero Dimensional Space|zero dimensional topological space]].
From [[Partition Topology is not T0|Partition Topology is not $T_0$]], $T$ is not a [[Definition:T0 Space... | Zero Dimensional Space is not necessarily T0 | https://proofwiki.org/wiki/Zero_Dimensional_Space_is_not_necessarily_T0 | https://proofwiki.org/wiki/Zero_Dimensional_Space_is_not_necessarily_T0 | [
"T0 Spaces",
"Zero Dimensional Spaces"
] | [
"Definition:Zero Dimensional Space",
"Definition:T0 Space"
] | [
"Definition:Partition Topology",
"Partition Topology is Zero Dimensional",
"Definition:Zero Dimensional Space",
"Partition Topology is not T0",
"Definition:T0 Space"
] |
proofwiki-13957 | Scattered Space is not necessarily T1 | Let $T = \struct {S, \tau}$ be a scattered topological space.
Then $T$ is not necessarily a $T_1$ space. | Let $T = \struct {S, \tau}$ be a non-trivial particular point space.
From Particular Point Space is Scattered, $T$ is a scattered space.
From Non-Trivial Particular Point Space is not $T_1$, $T$ is not a $T_1$ (Fréchet) space.
{{qed}} | Let $T = \struct {S, \tau}$ be a [[Definition:Scattered Space|scattered topological space]].
Then $T$ is not necessarily a [[Definition:T1 Space|$T_1$ space]]. | Let $T = \struct {S, \tau}$ be a [[Definition:Trivial Topological Space|non-trivial]] [[Definition:Particular Point Topology|particular point space]].
From [[Particular Point Space is Scattered]], $T$ is a [[Definition:Scattered Space|scattered space]].
From [[Non-Trivial Particular Point Space is not T1|Non-Trivial ... | Scattered Space is not necessarily T1 | https://proofwiki.org/wiki/Scattered_Space_is_not_necessarily_T1 | https://proofwiki.org/wiki/Scattered_Space_is_not_necessarily_T1 | [
"T1 Spaces",
"Scattered Spaces"
] | [
"Definition:Scattered Space",
"Definition:T1 Space"
] | [
"Definition:Trivial Topological Space",
"Definition:Particular Point Topology",
"Particular Point Space is Scattered",
"Definition:Scattered Space",
"Non-Trivial Particular Point Space is not T1",
"Definition:T1 Space"
] |
proofwiki-13958 | Existence of Connected Non-T1 Scattered Space | There exists at least one example of a connected topological space which is not a $T_1$ space, but which is also a scattered space. | Let $T$ be the divisor space.
From Divisor Space is Connected, $T$ is a connected space.
From Divisor Space is not $T_1$, $T$ is not a $T_1$ space.
From Divisor Space is Scattered, $T$ is a scattered space.
Hence the result.
{{qed}} | There exists at least one example of a [[Definition:Connected Topological Space|connected topological space]] which is not a [[Definition:T1 Space|$T_1$ space]], but which is also a [[Definition:Scattered Space|scattered space]]. | Let $T$ be the [[Definition:Divisor Topology|divisor space]].
From [[Divisor Space is Connected]], $T$ is a [[Definition:Connected Topological Space|connected space]].
From [[Divisor Space is not T1|Divisor Space is not $T_1$]], $T$ is not a [[Definition:T1 Space|$T_1$ space]].
From [[Divisor Space is Scattered]], ... | Existence of Connected Non-T1 Scattered Space | https://proofwiki.org/wiki/Existence_of_Connected_Non-T1_Scattered_Space | https://proofwiki.org/wiki/Existence_of_Connected_Non-T1_Scattered_Space | [
"T1 Spaces",
"Connected Topological Spaces",
"Scattered Spaces"
] | [
"Definition:Connected Topological Space",
"Definition:T1 Space",
"Definition:Scattered Space"
] | [
"Definition:Divisor Topology",
"Divisor Space is Connected",
"Definition:Connected Topological Space",
"Divisor Space is not T1",
"Definition:T1 Space",
"Divisor Space is Scattered",
"Definition:Scattered Space"
] |
proofwiki-13959 | Sorgenfrey Line is not Second-Countable | Let $T = \struct {\mathbb R, \tau}$ be the Sorgenfrey line.
Then $T$ is not second-countable. | {{Recall|Second-Countable Space|second-countable space}}
{{:Definition:Second-Countable Space}}
Suppose $\BB$ is a basis for $\tau$.
{{Recall|Analytic Basis|(analytic) basis}}
{{:Definition:Analytic Basis/Definition 2}}
Let $U \in \tau$ be arbitrary.
Then:
:$U = \hointr x {x + \epsilon}$
for some $x \in \R$ and $\epsil... | Let $T = \struct {\mathbb R, \tau}$ be the [[Definition:Sorgenfrey Line|Sorgenfrey line]].
Then $T$ is not [[Definition:Second-Countable Space|second-countable]]. | {{Recall|Second-Countable Space|second-countable space}}
{{:Definition:Second-Countable Space}}
Suppose $\BB$ is a [[Definition:Analytic Basis|basis]] for $\tau$.
{{Recall|Analytic Basis|(analytic) basis}}
{{:Definition:Analytic Basis/Definition 2}}
Let $U \in \tau$ be [[Definition:Arbitrary|arbitrary]].
Then:
:$U ... | Sorgenfrey Line is not Second-Countable | https://proofwiki.org/wiki/Sorgenfrey_Line_is_not_Second-Countable | https://proofwiki.org/wiki/Sorgenfrey_Line_is_not_Second-Countable | [
"Sorgenfrey Line",
"Examples of Second-Countable Spaces"
] | [
"Definition:Sorgenfrey Line",
"Definition:Second-Countable Space"
] | [
"Definition:Basis (Topology)/Analytic Basis",
"Definition:Arbitrary",
"Definition:Infimum of Set",
"Definition:Cardinality",
"Definition:Uncountable/Set"
] |
proofwiki-13960 | Biconnected Set does not necessarily have Dispersion Point | A biconnected set does not necessarily have a dispersion point. | Let $T$ be a Miller's biconnected set.
From Miller's Biconnected Set is Biconnected, $T$ is a biconnected set.
From Miller's Biconnected Set is has no Dispersion Point, $T$ does not have a dispersion point.
Hence the result.
{{qed}} | A [[Definition:Biconnected Set|biconnected set]] does not necessarily have a [[Definition:Dispersion Point|dispersion point]]. | Let $T$ be a [[Definition:Miller's Biconnected Set|Miller's biconnected set]].
From [[Miller's Biconnected Set is Biconnected]], $T$ is a [[Definition:Biconnected Set|biconnected set]].
From [[Miller's Biconnected Set is has no Dispersion Point]], $T$ does not have a [[Definition:Dispersion Point|dispersion point]]... | Biconnected Set does not necessarily have Dispersion Point | https://proofwiki.org/wiki/Biconnected_Set_does_not_necessarily_have_Dispersion_Point | https://proofwiki.org/wiki/Biconnected_Set_does_not_necessarily_have_Dispersion_Point | [
"Biconnected Sets",
"Dispersion Points"
] | [
"Definition:Biconnected Set",
"Definition:Dispersion Point"
] | [
"Definition:Miller's Biconnected Set",
"Miller's Biconnected Set is Biconnected",
"Definition:Biconnected Set",
"Miller's Biconnected Set is has no Dispersion Point",
"Definition:Dispersion Point"
] |
proofwiki-13961 | Existence of Connected Punctiform Space | There exists at least one example of a connected topological space which is also punctiform. | Let $T$ be Cantor's leaky tent.
From Cantor's Leaky Tent is Connected, $T$ is a connected topological space.
From Cantor's Leaky Tent is Punctiform, $T$ is a punctiform space.
Hence the result.
{{qed}} | There exists at least one example of a [[Definition:Connected Topological Space|connected topological space]] which is also [[Definition:Punctiform Space|punctiform]]. | Let $T$ be [[Definition:Cantor's Leaky Tent|Cantor's leaky tent]].
From [[Cantor's Leaky Tent is Connected]], $T$ is a [[Definition:Connected Topological Space|connected topological space]].
From [[Cantor's Leaky Tent is Punctiform]], $T$ is a [[Definition:Punctiform Space|punctiform space]].
Hence the result.
{{qe... | Existence of Connected Punctiform Space | https://proofwiki.org/wiki/Existence_of_Connected_Punctiform_Space | https://proofwiki.org/wiki/Existence_of_Connected_Punctiform_Space | [
"Connected Topological Spaces",
"Punctiform Spaces"
] | [
"Definition:Connected Topological Space",
"Definition:Punctiform Space"
] | [
"Definition:Cantor's Leaky Tent",
"Cantor's Leaky Tent is Connected",
"Definition:Connected Topological Space",
"Cantor's Leaky Tent is Punctiform",
"Definition:Punctiform Space"
] |
proofwiki-13962 | Metric Space is Completely Normal | A metric space $M = \struct {A, d}$ is a completely normal space. | {{Recall|Completely Normal Space|completely normal space|index = 1}}
{{:Definition:Completely Normal Space/Definition 1}}
We have that:
{{begin-itemize}}
{{item||a Metric Space is $T_5$}}
{{item||a Metric Space is $T_1$.}}
{{end-itemize}}
Hence the result.
{{qed}} | A [[Definition:Metric Space|metric space]] $M = \struct {A, d}$ is a [[Definition:Completely Normal Space|completely normal space]]. | {{Recall|Completely Normal Space|completely normal space|index = 1}}
{{:Definition:Completely Normal Space/Definition 1}}
We have that:
{{begin-itemize}}
{{item||a [[Metric Space is T5|Metric Space is $T_5$]]}}
{{item||a [[Metric Space is T1|Metric Space is $T_1$]].}}
{{end-itemize}}
Hence the result.
{{qed}} | Metric Space is Completely Normal | https://proofwiki.org/wiki/Metric_Space_is_Completely_Normal | https://proofwiki.org/wiki/Metric_Space_is_Completely_Normal | [
"Metric Space fulfils all Separation Axioms",
"Examples of Completely Normal Spaces"
] | [
"Definition:Metric Space",
"Definition:Completely Normal Space"
] | [
"Metric Space is T5",
"Metric Space is T1"
] |
proofwiki-13963 | Metric Space is Perfectly Normal | A metric space $M = \struct {A, d}$ is a perfectly normal space. | {{Recall|Perfectly Normal Space|perfectly normal space|index = 1}}
{{:Definition:Perfectly Normal Space/Definition 1}}
From Metric Space is Perfectly $T_4$
:$M$ is a perfectly $T_4$ space.
From Metric Space is $T_1$
:$M$ is a $T_1$ space.
The result follows.
{{qed}} | A [[Definition:Metric Space|metric space]] $M = \struct {A, d}$ is a [[Definition:Perfectly Normal Space|perfectly normal space]]. | {{Recall|Perfectly Normal Space|perfectly normal space|index = 1}}
{{:Definition:Perfectly Normal Space/Definition 1}}
From [[Metric Space is Perfectly T4|Metric Space is Perfectly $T_4$]]
:$M$ is a [[Definition:Perfectly T4 Space|perfectly $T_4$ space]].
From [[Metric Space is T1|Metric Space is $T_1$]]
:$M$ is a [[... | Metric Space is Perfectly Normal | https://proofwiki.org/wiki/Metric_Space_is_Perfectly_Normal | https://proofwiki.org/wiki/Metric_Space_is_Perfectly_Normal | [
"Metric Space fulfils all Separation Axioms",
"Examples of Perfectly Normal Spaces"
] | [
"Definition:Metric Space",
"Definition:Perfectly Normal Space"
] | [
"Metric Space is Perfectly T4",
"Definition:Perfectly T4 Space",
"Metric Space is T1",
"Definition:T1 Space"
] |
proofwiki-13964 | Bounded Metric Space is not necessarily Totally Bounded | Let $M = \struct {A, d}$ be a bounded metric space.
Then it is not necessarily the case that $M$ is totally bounded. | Let $d$ be a discrete metric on the open unit interval $\Bbb I := \openint 0 1 \subseteq \R$.
We have that for all $x \in \openint 0 1$ and for all $r \in \R_{> 1}$:
:$\map {B_r} x = \openint 0 1$
where $\map {B_r} x$ denotes the open $r$-ball of $x$.
Thus $\struct {\Bbb I, d}$ is bounded.
Let $\epsilon \in \R_{>0}$ be... | Let $M = \struct {A, d}$ be a [[Definition:Bounded Metric Space|bounded metric space]].
Then it is not necessarily the case that $M$ is [[Definition:Totally Bounded Metric Space|totally bounded]]. | Let $d$ be a [[Definition:Standard Discrete Metric|discrete metric]] on the [[Definition:Open Unit Interval|open unit interval]] $\Bbb I := \openint 0 1 \subseteq \R$.
We have that for all $x \in \openint 0 1$ and for all $r \in \R_{> 1}$:
:$\map {B_r} x = \openint 0 1$
where $\map {B_r} x$ denotes the [[Definition:Op... | Bounded Metric Space is not necessarily Totally Bounded/Proof 2 | https://proofwiki.org/wiki/Bounded_Metric_Space_is_not_necessarily_Totally_Bounded | https://proofwiki.org/wiki/Bounded_Metric_Space_is_not_necessarily_Totally_Bounded/Proof_2 | [
"Bounded Metric Space is not necessarily Totally Bounded",
"Bounded Metric Spaces",
"Totally Bounded Metric Spaces"
] | [
"Definition:Bounded Metric Space",
"Definition:Totally Bounded Metric Space"
] | [
"Definition:Standard Discrete Metric",
"Definition:Real Interval/Unit Interval/Open",
"Definition:Open Ball",
"Definition:Bounded Metric Space",
"Definition:Strictly Positive/Real Number",
"Definition:Epsilon-Net/Finite Net",
"Definition:Totally Bounded Metric Space",
"Definition:Totally Bounded Metri... |
proofwiki-13965 | Total Boundedness is not Preserved under Homeomorphism | Let $M = \struct {A, d}$ be a totally bounded metric space.
Let $M' = \struct {A', d'}$ be a metric space.
Let $M$ be homeomorphic to $M'$.
Then it is not necessarily the case that $M'$ is totally bounded. | {{ProofWanted|According to S&S (item $134$), this is proved somehow using the metric $\delta {{=}} \dfrac d {1 + d}$, given some metric space $M {{=}} \struct {A, d}$, but the derivation of this is obscure.}} | Let $M = \struct {A, d}$ be a [[Definition:Totally Bounded Metric Space|totally bounded metric space]].
Let $M' = \struct {A', d'}$ be a [[Definition:Metric Space|metric space]].
Let $M$ be [[Definition:Homeomorphic Metric Spaces|homeomorphic]] to $M'$.
Then it is not necessarily the case that $M'$ is [[Definition:... | {{ProofWanted|According to S&S (item $134$), this is proved somehow using the metric $\delta {{=}} \dfrac d {1 + d}$, given some [[Definition:Metric Space|metric space]] $M {{=}} \struct {A, d}$, but the derivation of this is obscure.}} | Total Boundedness is not Preserved under Homeomorphism | https://proofwiki.org/wiki/Total_Boundedness_is_not_Preserved_under_Homeomorphism | https://proofwiki.org/wiki/Total_Boundedness_is_not_Preserved_under_Homeomorphism | [
"Homeomorphisms (Metric Spaces)",
"Totally Bounded Metric Spaces"
] | [
"Definition:Totally Bounded Metric Space",
"Definition:Metric Space",
"Definition:Homeomorphism/Metric Spaces",
"Definition:Totally Bounded Metric Space"
] | [
"Definition:Metric Space"
] |
proofwiki-13966 | Tutte's Wheel Theorem | Every $3$-connected graph can be obtained by the following procedure:
* Start with $G_0 := K_4$
* Given $G_i$ pick a vertex $v$
* Split into $v'$ and $v' '$ and add edge $\set {v', v' '}$
This procedure directly follows from the theorem:
:A graph $G$ is $3$-connected ('''A''') {{iff}} there exists a sequence $G_0, G_1... | === Lemma ===
{{:Tutte's Wheel Theorem/Lemma}}{{qed|lemma}} | Every [[Definition:K-Connected|$3$-connected]] [[Definition:Graph (Graph Theory)|graph]] can be obtained by the following procedure:
* Start with $G_0 := K_4$
* Given $G_i$ pick a vertex $v$
* Split into $v'$ and $v' '$ and add edge $\set {v', v' '}$
This procedure directly follows from the theorem:
:A graph $G$ is... | === [[Tutte's Wheel Theorem/Lemma|Lemma]] ===
{{:Tutte's Wheel Theorem/Lemma}}{{qed|lemma}} | Tutte's Wheel Theorem | https://proofwiki.org/wiki/Tutte's_Wheel_Theorem | https://proofwiki.org/wiki/Tutte's_Wheel_Theorem | [
"Tutte's Wheel Theorem",
"Graph Theory"
] | [
"Definition:K-Connected",
"Definition:Graph (Graph Theory)",
"Definition:K-Connected"
] | [
"Tutte's Wheel Theorem/Lemma"
] |
proofwiki-13967 | Complete Metrizability is not Hereditary | Let $T = \struct {S, \tau}$ be a topological space which is completely metrizable.
Let $H \subseteq S$ be a subset of $S$.
Let $\struct {H, \tau_H}$ be the topological subspace of $T$ induced by $H$.
Then it is not necessarily the case that $\struct {H, \tau_H}$ is also completely metrizable.
That is, complete metrizab... | Let $\struct {\R, d}$ denote the real number line under the Euclidean metric.
Let $\struct {\Q, d}$ denote the rational number space under the Euclidean metric.
We have that $\Q \subset \R$ by definition.
Let $T = \struct {\R, \tau_d}$ be the topological space induced on $\R$ by $d$.
By Real Number Line is Complete Met... | Let $T = \struct {S, \tau}$ be a [[Definition:Topological Space|topological space]] which is [[Definition:Completely Metrizable Space|completely metrizable]].
Let $H \subseteq S$ be a [[Definition:Subset|subset]] of $S$.
Let $\struct {H, \tau_H}$ be the [[Definition:Topological Subspace|topological subspace]] of $T$ ... | Let $\struct {\R, d}$ denote the [[Definition:Real Number Line|real number line]] under the [[Definition:Euclidean Metric on Real Number Line|Euclidean metric]].
Let $\struct {\Q, d}$ denote the [[Definition:Rational Number Space|rational number space]] under the [[Definition:Euclidean Metric on Real Number Line|Eucli... | Complete Metrizability is not Hereditary | https://proofwiki.org/wiki/Complete_Metrizability_is_not_Hereditary | https://proofwiki.org/wiki/Complete_Metrizability_is_not_Hereditary | [
"Completely Metrizable Spaces",
"Examples of Hereditary Properties"
] | [
"Definition:Topological Space",
"Definition:Completely Metrizable Space",
"Definition:Subset",
"Definition:Topological Subspace",
"Definition:Completely Metrizable Space",
"Definition:Completely Metrizable Space",
"Definition:Hereditary Property (Topology)"
] | [
"Definition:Real Number/Real Number Line",
"Definition:Euclidean Metric/Real Number Line",
"Definition:Rational Number Space",
"Definition:Euclidean Metric/Real Number Line",
"Definition:Topology Induced by Metric",
"Real Number Line is Complete Metric Space",
"Definition:Complete Metric Space",
"Defi... |
proofwiki-13968 | Completion Theorem (Metric Space)/Lemma 4 | :$\tilde M = \struct {\tilde A, \tilde d}$ is unique up to isometry. | Let $M_1 = \struct {\tilde{A_1}, \tilde{d_1}, \phi_1}$ and $M_2 = \struct {\tilde{A_2}, \tilde{d_2}, \phi_2}$ be two completions of $\struct {A, d}$.
Here, $\phi_1: A \to A_1$ and $\phi_2: A \to A_2$ are isometries
By Composite of Isometries is Isometry, $\psi = \phi_2 \circ \phi_1^{-1}$ gives an isometry from $\phi_1 ... | :$\tilde M = \struct {\tilde A, \tilde d}$ is unique up to [[Definition:Isometry (Metric Spaces)|isometry]]. | Let $M_1 = \struct {\tilde{A_1}, \tilde{d_1}, \phi_1}$ and $M_2 = \struct {\tilde{A_2}, \tilde{d_2}, \phi_2}$ be two [[Definition:Completion of Metric Space|completions]] of $\struct {A, d}$.
Here, $\phi_1: A \to A_1$ and $\phi_2: A \to A_2$ are [[Definition:Isometry (Metric Spaces)|isometries]]
By [[Composite of Is... | Completion Theorem (Metric Space)/Lemma 4 | https://proofwiki.org/wiki/Completion_Theorem_(Metric_Space)/Lemma_4 | https://proofwiki.org/wiki/Completion_Theorem_(Metric_Space)/Lemma_4 | [
"Completion Theorem (Metric Space)"
] | [
"Definition:Isometry (Metric Spaces)"
] | [
"Definition:Completion (Metric Space)",
"Definition:Isometry (Metric Spaces)",
"Composite of Isometries is Isometry",
"Definition:Isometry (Metric Spaces)",
"Definition:Set",
"Definition:Everywhere Dense",
"Definition:Extension of Mapping",
"Definition:Continuous Mapping (Metric Space)",
"Definition... |
proofwiki-13969 | Sorgenfrey Line is First-Countable | Let $\R$ be the set of real numbers.
Let $\BB = \set {\hointr a b: a, b \in \R}$.
Let $\tau$ be the topology generated by $\BB$, that is, the Sorgenfrey line.
Then $\tau$ is first-countable. | {{Recall|First-Countable Space|first-countable space}}
{{:Definition:First-Countable Space}}
Let $\BB_x = \set {\hointr x {x + \dfrac 1 n} : n \in \N_{>0} }$.
We will show that:
:$(1): \quad \BB_x$ is countable
:$(2): \quad \BB_x$ is a local basis at $x$
$(1)$ follows from the fact that $\BB_x$ is a bijection from the ... | Let $\R$ be the [[Definition:Real Number|set of real numbers]].
Let $\BB = \set {\hointr a b: a, b \in \R}$.
Let $\tau$ be the [[Definition:Topology|topology]] [[Definition:Topology Generated by Synthetic Basis|generated]] by $\BB$, that is, the [[Definition:Sorgenfrey Line|Sorgenfrey line]].
Then $\tau$ is [[Defin... | {{Recall|First-Countable Space|first-countable space}}
{{:Definition:First-Countable Space}}
Let $\BB_x = \set {\hointr x {x + \dfrac 1 n} : n \in \N_{>0} }$.
We will show that:
:$(1): \quad \BB_x$ is [[Definition:Countable Set|countable]]
:$(2): \quad \BB_x$ is a [[Definition:Local Basis|local basis]] at $x$
$(1)... | Sorgenfrey Line is First-Countable | https://proofwiki.org/wiki/Sorgenfrey_Line_is_First-Countable | https://proofwiki.org/wiki/Sorgenfrey_Line_is_First-Countable | [
"Sorgenfrey Line",
"Examples of First-Countable Spaces"
] | [
"Definition:Real Number",
"Definition:Topology",
"Definition:Topology Generated by Synthetic Basis",
"Definition:Sorgenfrey Line",
"Definition:First-Countable Space"
] | [
"Definition:Countable Set",
"Definition:Local Basis",
"Definition:Bijection",
"Definition:Natural Numbers",
"Definition:Local Basis",
"Definition:Arbitrary",
"Axiom of Archimedes",
"Definition:Local Basis",
"Definition:Sorgenfrey Line",
"Definition:Countable Set",
"Definition:Local Basis",
"De... |
proofwiki-13970 | Metrizable Space is not necessarily Second-Countable | Let $T = \struct {S, \tau}$ be a topological space which is metrizable.
Then it is not necessarily the case that $T$ is second-countable. | Let $T$ be an uncountable discrete space.
From Standard Discrete Metric induces Discrete Topology, $T$ is metrizable.
From Uncountable Discrete Space is not Second-Countable, $T$ is not second-countable.
{{qed}} | Let $T = \struct {S, \tau}$ be a [[Definition:Topological Space|topological space]] which is [[Definition:Metrizable Space|metrizable]].
Then it is not necessarily the case that $T$ is [[Definition:Second-Countable Space|second-countable]]. | Let $T$ be an [[Definition:Uncountable Discrete Topology|uncountable discrete space]].
From [[Standard Discrete Metric induces Discrete Topology]], $T$ is [[Definition:Metrizable Space|metrizable]].
From [[Uncountable Discrete Space is not Second-Countable]], $T$ is not [[Definition:Second-Countable Space|second-coun... | Metrizable Space is not necessarily Second-Countable | https://proofwiki.org/wiki/Metrizable_Space_is_not_necessarily_Second-Countable | https://proofwiki.org/wiki/Metrizable_Space_is_not_necessarily_Second-Countable | [
"Metrizable Spaces",
"Second-Countable Spaces"
] | [
"Definition:Topological Space",
"Definition:Metrizable Space",
"Definition:Second-Countable Space"
] | [
"Definition:Discrete Topology/Uncountable",
"Standard Discrete Metric induces Discrete Topology",
"Definition:Metrizable Space",
"Uncountable Discrete Space is not Second-Countable",
"Definition:Second-Countable Space"
] |
proofwiki-13971 | Regular Paracompact Space is not necessarily Metrizable | Let $T = \struct {S, \tau}$ be a topological space which is regular and paracompact.
Then it is not necessarily the case that $T$ is metrizable. | Let $T$ be the Sorgenfrey line.
From Sorgenfrey Line satisfies all Separation Axioms, $T$ is a regular space.
From Sorgenfrey Line is Paracompact, $T$ is a paracompact space.
From Sorgenfrey Line is not Metrizable, $T$ is not a metrizable space.
{{qed}} | Let $T = \struct {S, \tau}$ be a [[Definition:Topological Space|topological space]] which is [[Definition:Regular Space|regular]] and [[Definition:Paracompact Space|paracompact]].
Then it is not necessarily the case that $T$ is [[Definition:Metrizable Space|metrizable]]. | Let $T$ be the [[Definition:Sorgenfrey Line|Sorgenfrey line]].
From [[Sorgenfrey Line satisfies all Separation Axioms]], $T$ is a [[Definition:Regular Space|regular space]].
From [[Sorgenfrey Line is Paracompact]], $T$ is a [[Definition:Paracompact Space|paracompact space]].
From [[Sorgenfrey Line is not Metrizable]... | Regular Paracompact Space is not necessarily Metrizable/Proof 1 | https://proofwiki.org/wiki/Regular_Paracompact_Space_is_not_necessarily_Metrizable | https://proofwiki.org/wiki/Regular_Paracompact_Space_is_not_necessarily_Metrizable/Proof_1 | [
"Regular Paracompact Space is not necessarily Metrizable",
"Metrizable Spaces",
"Paracompact Spaces",
"Regular Spaces"
] | [
"Definition:Topological Space",
"Definition:Regular Space",
"Definition:Paracompact Space",
"Definition:Metrizable Space"
] | [
"Definition:Sorgenfrey Line",
"Sorgenfrey Line satisfies all Separation Axioms",
"Definition:Regular Space",
"Sorgenfrey Line is Paracompact",
"Definition:Paracompact Space",
"Sorgenfrey Line is not Metrizable",
"Definition:Metrizable Space"
] |
proofwiki-13972 | Regular Paracompact Space is not necessarily Metrizable | Let $T = \struct {S, \tau}$ be a topological space which is regular and paracompact.
Then it is not necessarily the case that $T$ is metrizable. | Let $T$ be the radial interval space.
From Radial Interval Space is Completely Normal, $T$ is a completely normal space.
Hence from Sequence of Implications of Separation Axioms, $T$ is a regular space.
From Radial Interval Space is Paracompact, $T$ is a paracompact space.
From Radial Interval Space is not Metrizable, ... | Let $T = \struct {S, \tau}$ be a [[Definition:Topological Space|topological space]] which is [[Definition:Regular Space|regular]] and [[Definition:Paracompact Space|paracompact]].
Then it is not necessarily the case that $T$ is [[Definition:Metrizable Space|metrizable]]. | Let $T$ be the [[Definition:Radial Interval Space|radial interval space]].
From [[Radial Interval Space is Completely Normal]], $T$ is a [[Definition:Completely Normal Space|completely normal space]].
Hence from [[Sequence of Implications of Separation Axioms]], $T$ is a [[Definition:Regular Space|regular space]].
F... | Regular Paracompact Space is not necessarily Metrizable/Proof 2 | https://proofwiki.org/wiki/Regular_Paracompact_Space_is_not_necessarily_Metrizable | https://proofwiki.org/wiki/Regular_Paracompact_Space_is_not_necessarily_Metrizable/Proof_2 | [
"Regular Paracompact Space is not necessarily Metrizable",
"Metrizable Spaces",
"Paracompact Spaces",
"Regular Spaces"
] | [
"Definition:Topological Space",
"Definition:Regular Space",
"Definition:Paracompact Space",
"Definition:Metrizable Space"
] | [
"Definition:Radial Interval Topology",
"Radial Interval Space is Completely Normal",
"Definition:Completely Normal Space",
"Sequence of Implications of Separation Axioms",
"Definition:Regular Space",
"Radial Interval Space is Paracompact",
"Definition:Paracompact Space",
"Radial Interval Space is not ... |
proofwiki-13973 | Uniform Space whose Topology is Metrizable is not necessarily Metrizable | Let $\UU$ be a uniformity on a set $S$.
Let $\struct {\struct {S, \UU}, \tau}$ be the uniform space generated from $\UU$.
Let $T = \struct {S, \tau}$ be the uniformizable space yielded by $\struct {\struct {S, \UU}, \tau}$.
Let $T$ be a metrizable space.
Then it is not necessarily the case that $\UU$ is itself a metriz... | Let $T = \struct {S, \tau}$ be an uncountable discrete ordinal space.
From Uncountable Discrete Ordinal Space is Metrizable, $T$ is a metrizable space.
However, from Uncountable Discrete Ordinal Space has Unmetrizable Uniformity, there exists a uniformity $\UU$ which yields the uniformizable space $T = \struct {S, \tau... | Let $\UU$ be a [[Definition:Uniformity|uniformity]] on a [[Definition:Set|set]] $S$.
Let $\struct {\struct {S, \UU}, \tau}$ be the [[Definition:Uniform Space|uniform space]] generated from $\UU$.
Let $T = \struct {S, \tau}$ be the [[Definition:Uniformizable Space|uniformizable space]] yielded by $\struct {\struct {S,... | Let $T = \struct {S, \tau}$ be an [[Definition:Uncountable Discrete Ordinal Space|uncountable discrete ordinal space]].
From [[Uncountable Discrete Ordinal Space is Metrizable]], $T$ is a [[Definition:Metrizable Space|metrizable space]].
However, from [[Uncountable Discrete Ordinal Space has Unmetrizable Uniformity]]... | Uniform Space whose Topology is Metrizable is not necessarily Metrizable | https://proofwiki.org/wiki/Uniform_Space_whose_Topology_is_Metrizable_is_not_necessarily_Metrizable | https://proofwiki.org/wiki/Uniform_Space_whose_Topology_is_Metrizable_is_not_necessarily_Metrizable | [
"Metrizable Spaces",
"Uniformities"
] | [
"Definition:Uniformity",
"Definition:Set",
"Definition:Uniform Space",
"Definition:T3.5 Space",
"Definition:Metrizable Space",
"Definition:Metrizable Uniformity"
] | [
"Definition:Uncountable Discrete Ordinal Space",
"Uncountable Discrete Ordinal Space is Metrizable",
"Definition:Metrizable Space",
"Uncountable Discrete Ordinal Space has Unmetrizable Uniformity",
"Definition:Uniformity",
"Definition:T3.5 Space",
"Definition:Metrizable Uniformity"
] |
proofwiki-13974 | Open Sets in Indiscrete Topology | $H$ is an open set of $T$ {{iff}} either $H = S$ or $H = \O$. | A set $U$ is open in a topology $\tau$ if $U \in \tau$.
In $\tau = \set {\O, S}$, the only open sets are $\O$ and $S$.
{{qed}} | $H$ is an [[Definition:Open Set (Topology)|open set]] of $T$ {{iff}} either $H = S$ or $H = \O$. | A set $U$ is [[Definition:Open Set (Topology)|open]] in a [[Definition:Topology|topology]] $\tau$ if $U \in \tau$.
In $\tau = \set {\O, S}$, the only [[Definition:Open Set (Topology)|open sets]] are $\O$ and $S$.
{{qed}} | Open Sets in Indiscrete Topology | https://proofwiki.org/wiki/Open_Sets_in_Indiscrete_Topology | https://proofwiki.org/wiki/Open_Sets_in_Indiscrete_Topology | [
"Indiscrete Topology",
"Examples of Open Sets"
] | [
"Definition:Open Set/Topology"
] | [
"Definition:Open Set/Topology",
"Definition:Topology",
"Definition:Open Set/Topology"
] |
proofwiki-13975 | Closed Sets in Indiscrete Topology | $H$ is a closed set of $T$ {{iff}} either $H = S$ or $H = \O$. | A set $U$ is closed in a topology $\tau$ {{iff}}:
:$\relcomp S U \in \tau$
where $\relcomp S U$ denotes the complement of $U$ in $S$.
That is, the closed sets are the complements of the open sets.
From Open Sets in Indiscrete Topology, in $\tau = \set {\O, S}$, the only open sets are $\O$ and $S$.
Hence the only closed... | $H$ is a [[Definition:Closed Set (Topology)|closed set]] of $T$ {{iff}} either $H = S$ or $H = \O$. | A set $U$ is [[Definition:Closed Set (Topology)|closed]] in a [[Definition:Topology|topology]] $\tau$ {{iff}}:
:$\relcomp S U \in \tau$
where $\relcomp S U$ denotes the [[Definition:Relative Complement|complement]] of $U$ in $S$.
That is, the [[Definition:Closed Set (Topology)|closed sets]] are the [[Definition:Relati... | Closed Sets in Indiscrete Topology | https://proofwiki.org/wiki/Closed_Sets_in_Indiscrete_Topology | https://proofwiki.org/wiki/Closed_Sets_in_Indiscrete_Topology | [
"Indiscrete Topology",
"Examples of Closed Sets"
] | [
"Definition:Closed Set/Topology"
] | [
"Definition:Closed Set/Topology",
"Definition:Topology",
"Definition:Relative Complement",
"Definition:Closed Set/Topology",
"Definition:Relative Complement",
"Definition:Open Set/Topology",
"Open Sets in Indiscrete Topology",
"Definition:Open Set/Topology",
"Definition:Closed Set/Topology",
"Defi... |
proofwiki-13976 | F-Sigma Sets in Indiscrete Topology | $H \subseteq S$ is an $F_\sigma$ ($F$-sigma) set of an indiscrete topological space $T = \struct {S, \set {\O, S} }$ {{iff}} either $H = S$ or $H = \O$. | An $F_\sigma$ set is a set which can be written as a countable union of closed sets of $S$.
Hence the only $F_\sigma$ sets of $T$ are made from unions of $T$ and $\O$.
So $T$ and $\O$ are the only $F_\sigma$ sets of $T$.
{{qed}} | $H \subseteq S$ is an [[Definition:F-Sigma Set|$F_\sigma$ ($F$-sigma) set]] of an [[Definition:Indiscrete Space|indiscrete topological space]] $T = \struct {S, \set {\O, S} }$ {{iff}} either $H = S$ or $H = \O$. | An [[Definition:F-Sigma Set|$F_\sigma$ set]] is a [[Definition:Set|set]] which can be written as a [[Definition:Countable Union|countable union]] of [[Definition:Closed Set (Topology)|closed sets]] of $S$.
Hence the only [[Definition:F-Sigma Set|$F_\sigma$ sets]] of $T$ are made from [[Definition:Set Union|unions]] of... | F-Sigma Sets in Indiscrete Topology | https://proofwiki.org/wiki/F-Sigma_Sets_in_Indiscrete_Topology | https://proofwiki.org/wiki/F-Sigma_Sets_in_Indiscrete_Topology | [
"Indiscrete Topology",
"Examples of F-Sigma Sets"
] | [
"Definition:F-Sigma Set",
"Definition:Indiscrete Topology"
] | [
"Definition:F-Sigma Set",
"Definition:Set",
"Definition:Set Union/Countable Union",
"Definition:Closed Set/Topology",
"Definition:F-Sigma Set",
"Definition:Set Union",
"Definition:F-Sigma Set"
] |
proofwiki-13977 | G-Delta Sets in Indiscrete Topology | $H \subseteq S$ is a $G_\delta$ ($G$-delta) set of an indiscrete topological space $T = \struct {S, \set {\O, S} }$ {{iff}} either $H = S$ or $H = \O$. | A $G_\delta$ set is a set which can be written as a countable intersection of open sets of $S$.
Hence the only $G_\delta$ sets of $T$ are made from intersections of $T$ and $\O$.
So $T$ and $\O$ are the only $G_\delta$ sets of $T$.
{{qed}} | $H \subseteq S$ is a [[Definition:G-Delta Set|$G_\delta$ ($G$-delta) set]] of an [[Definition:Indiscrete Space|indiscrete topological space]] $T = \struct {S, \set {\O, S} }$ {{iff}} either $H = S$ or $H = \O$. | A [[Definition:G-Delta Set|$G_\delta$ set]] is a [[Definition:Set|set]] which can be written as a [[Definition:Countable Intersection|countable intersection]] of [[Definition:Open Set (Topology)|open sets]] of $S$.
Hence the only [[Definition:G-Delta Set|$G_\delta$ sets]] of $T$ are made from [[Definition:Set Intersec... | G-Delta Sets in Indiscrete Topology | https://proofwiki.org/wiki/G-Delta_Sets_in_Indiscrete_Topology | https://proofwiki.org/wiki/G-Delta_Sets_in_Indiscrete_Topology | [
"Indiscrete Topology",
"Examples of G-Delta Sets"
] | [
"Definition:G-Delta Set",
"Definition:Indiscrete Topology"
] | [
"Definition:G-Delta Set",
"Definition:Set",
"Definition:Set Intersection/Countable Intersection",
"Definition:Open Set/Topology",
"Definition:G-Delta Set",
"Definition:Set Intersection",
"Definition:G-Delta Set"
] |
proofwiki-13978 | Subset of Indiscrete Space is Compact | $H \subseteq S$ is compact in an indiscrete topological space $T = \struct {S, \set {\O, S} }$. | The subspace $T_H = \struct {H, \set {\O, S \cap H} }$ is trivially also an indiscrete space.
The only open cover of $T_H$ is $\set H$ itself.
The only subcover of $H$ is, trivially, also $\set H$, which is finite.
So $H$ is (trivially) compact in $T$.
{{qed}} | $H \subseteq S$ is [[Definition:Compact Topological Subspace|compact]] in an [[Definition:Indiscrete Space|indiscrete topological space]] $T = \struct {S, \set {\O, S} }$. | The [[Definition:Topological Subspace|subspace]] $T_H = \struct {H, \set {\O, S \cap H} }$ is trivially also an [[Definition:Indiscrete Space|indiscrete space]].
The only [[Definition:Open Cover|open cover]] of $T_H$ is $\set H$ itself.
The only [[Definition:Subcover|subcover]] of $H$ is, trivially, also $\set H$, wh... | Subset of Indiscrete Space is Compact | https://proofwiki.org/wiki/Subset_of_Indiscrete_Space_is_Compact | https://proofwiki.org/wiki/Subset_of_Indiscrete_Space_is_Compact | [
"Indiscrete Topology",
"Examples of Compact Topological Spaces"
] | [
"Definition:Compact Topological Space/Subspace",
"Definition:Indiscrete Topology"
] | [
"Definition:Topological Subspace",
"Definition:Indiscrete Topology",
"Definition:Open Cover",
"Definition:Subcover",
"Definition:Subcover/Finite",
"Definition:Compact Topological Space/Subspace"
] |
proofwiki-13979 | Subset of Indiscrete Space is Sequentially Compact | $H \subseteq S$ is sequentially compact in an indiscrete topological space $T = \struct {S, \set {\O, S} }$. | From Sequence in Indiscrete Space converges to Every Point, every sequence in $T$ converges to every point of $S$.
So every infinite sequence has a subsequence which converges to every point in $S$.
Hence $H$ is (trivially) sequentially compact in $T$.
{{qed}} | $H \subseteq S$ is [[Definition:Sequentially Compact Space|sequentially compact]] in an [[Definition:Indiscrete Space|indiscrete topological space]] $T = \struct {S, \set {\O, S} }$. | From [[Sequence in Indiscrete Space converges to Every Point]], every [[Definition:Sequence|sequence]] in $T$ [[Definition:Convergent Sequence (Topology)|converges]] to every point of $S$.
So every [[Definition:Infinite Sequence|infinite sequence]] has a [[Definition:Subsequence|subsequence]] which [[Definition:Conver... | Subset of Indiscrete Space is Sequentially Compact | https://proofwiki.org/wiki/Subset_of_Indiscrete_Space_is_Sequentially_Compact | https://proofwiki.org/wiki/Subset_of_Indiscrete_Space_is_Sequentially_Compact | [
"Indiscrete Topology",
"Examples of Sequentially Compact Spaces"
] | [
"Definition:Sequentially Compact Space",
"Definition:Indiscrete Topology"
] | [
"Sequence in Indiscrete Space converges to Every Point",
"Definition:Sequence",
"Definition:Convergent Sequence/Topology",
"Definition:Sequence/Infinite Sequence",
"Definition:Subsequence",
"Definition:Convergent Sequence/Topology",
"Definition:Sequentially Compact Space"
] |
proofwiki-13980 | Limit Points of Sequence in Indiscrete Space on Uncountable Set | Let $S$ be an uncountable set.
Let $T = \struct {S, \set {\O, S} }$ be the indiscrete topological space on $S$.
Let $\sequence {s_n}$ be a sequence in $T$.
Then every sequence in $T$ has an uncountable number of limit points. | From Sequence in Indiscrete Space converges to Every Point, $\sequence {s_n}$ converges to every point of $S$.
As $S$ is uncountable, the result follows.
{{qed}} | Let $S$ be an [[Definition:Uncountable Set|uncountable set]].
Let $T = \struct {S, \set {\O, S} }$ be the [[Definition:Indiscrete Space|indiscrete topological space]] on $S$.
Let $\sequence {s_n}$ be a [[Definition:Sequence|sequence]] in $T$.
Then every [[Definition:Sequence|sequence]] in $T$ has an [[Definition:Un... | From [[Sequence in Indiscrete Space converges to Every Point]], $\sequence {s_n}$ [[Definition:Convergent Sequence (Topology)|converges]] to every point of $S$.
As $S$ is [[Definition:Uncountable Set|uncountable]], the result follows.
{{qed}} | Limit Points of Sequence in Indiscrete Space on Uncountable Set | https://proofwiki.org/wiki/Limit_Points_of_Sequence_in_Indiscrete_Space_on_Uncountable_Set | https://proofwiki.org/wiki/Limit_Points_of_Sequence_in_Indiscrete_Space_on_Uncountable_Set | [
"Indiscrete Topology",
"Examples of Limit Points of Sequences"
] | [
"Definition:Uncountable/Set",
"Definition:Indiscrete Topology",
"Definition:Sequence",
"Definition:Sequence",
"Definition:Uncountable/Set",
"Definition:Limit of Sequence/Topological Space"
] | [
"Sequence in Indiscrete Space converges to Every Point",
"Definition:Convergent Sequence/Topology",
"Definition:Uncountable/Set"
] |
proofwiki-13981 | Partition Topology is not T2.5 | Let $S$ be a set and let $\PP$ be a partition on $S$ which is not the (trivial) partition of singletons.
Let $T = \struct {S, \tau}$ be the partition space whose basis is $\PP$.
Then $T$ is not a $T_{2 \frac 1 2}$ space. | {{AimForCont}} $T$ is a $T_{2 \frac 1 2}$ space.
Then from $T_{2 \frac 1 2}$ Space is $T_2$, $T$ is a $T_2$ space.
This contradicts the result Partition Space is not $T_2$.
Hence the result, by Proof by Contradiction.
{{qed}} | Let $S$ be a [[Definition:Set|set]] and let $\PP$ be a [[Definition:Partition (Set Theory)|partition]] on $S$ which is not the [[Definition:Partition of Singletons|(trivial) partition of singletons]].
Let $T = \struct {S, \tau}$ be the [[Definition:Partition Space|partition space]] whose [[Basis for Partition Topology... | {{AimForCont}} $T$ is a [[Definition:T2.5 Space|$T_{2 \frac 1 2}$ space]].
Then from [[T2.5 Space is T2|$T_{2 \frac 1 2}$ Space is $T_2$]], $T$ is a [[Definition:T2 Space|$T_2$ space]].
This [[Definition:Contradiction|contradicts]] the result [[Partition Space is not T2|Partition Space is not $T_2$]].
Hence the resu... | Partition Topology is not T2.5 | https://proofwiki.org/wiki/Partition_Topology_is_not_T2.5 | https://proofwiki.org/wiki/Partition_Topology_is_not_T2.5 | [
"Partition Topologies",
"Examples of T2.5 Spaces"
] | [
"Definition:Set",
"Definition:Set Partition",
"Definition:Trivial Partition/Partition of Singletons",
"Definition:Partition Topology",
"Basis for Partition Topology",
"Definition:T2.5 Space"
] | [
"Definition:T2.5 Space",
"T2.5 Space is T2",
"Definition:T2 Space",
"Definition:Contradiction",
"Partition Space is not T2",
"Proof by Contradiction"
] |
proofwiki-13982 | Partition Space is not T2 | Let $S$ be a set and let $\PP$ be a partition on $S$ which is not the (trivial) partition of singletons.
Let $T = \struct {S, \tau}$ be the partition space whose basis is $\PP$.
Then $T$ is not a $T_2$ (Hausdorff) space. | {{AimForCont}} $T$ is a $T_2$ (Hausdorff) space.
Then from $T_2$ Space is $T_1$, $T$ is a $T_1$ space.
This contradicts the result Partition Space is not $T_1$.
Hence the result, by Proof by Contradiction.
{{qed}} | Let $S$ be a [[Definition:Set|set]] and let $\PP$ be a [[Definition:Partition (Set Theory)|partition]] on $S$ which is not the [[Definition:Partition of Singletons|(trivial) partition of singletons]].
Let $T = \struct {S, \tau}$ be the [[Definition:Partition Space|partition space]] whose [[Basis for Partition Topology... | {{AimForCont}} $T$ is a [[Definition:T2 Space|$T_2$ (Hausdorff) space]].
Then from [[T2 Space is T1|$T_2$ Space is $T_1$]], $T$ is a [[Definition:T1 Space|$T_1$ space]].
This [[Definition:Contradiction|contradicts]] the result [[Partition Space is not T1|Partition Space is not $T_1$]].
Hence the result, by [[Proof b... | Partition Space is not T2 | https://proofwiki.org/wiki/Partition_Space_is_not_T2 | https://proofwiki.org/wiki/Partition_Space_is_not_T2 | [
"Partition Topologies",
"Examples of Hausdorff Spaces"
] | [
"Definition:Set",
"Definition:Set Partition",
"Definition:Trivial Partition/Partition of Singletons",
"Definition:Partition Topology",
"Basis for Partition Topology",
"Definition:T2 Space"
] | [
"Definition:T2 Space",
"T2 Space is T1",
"Definition:T1 Space",
"Definition:Contradiction",
"Partition Space is not T1",
"Proof by Contradiction"
] |
proofwiki-13983 | Partition Space is not T1 | Let $S$ be a set and let $\PP$ be a partition on $S$ which is not the (trivial) partition of singletons.
Let $T = \struct {S, \tau}$ be the partition space whose basis is $\PP$.
Then $T$ is not a $T_1$ space. | {{AimForCont}} $T$ is a $T_1$ space.
Then from $T_1$ Space is $T_0$ Space, $T$ is a $T_0$ space.
This contradicts the result Partition Topology is not $T_0$.
Hence the result, by Proof by Contradiction.
{{qed}} | Let $S$ be a [[Definition:Set|set]] and let $\PP$ be a [[Definition:Partition (Set Theory)|partition]] on $S$ which is not the [[Definition:Partition of Singletons|(trivial) partition of singletons]].
Let $T = \struct {S, \tau}$ be the [[Definition:Partition Space|partition space]] whose [[Basis for Partition Topology... | {{AimForCont}} $T$ is a [[Definition:T1 Space|$T_1$ space]].
Then from [[T1 Space is T0 Space|$T_1$ Space is $T_0$ Space]], $T$ is a [[Definition:T0 Space|$T_0$ space]].
This [[Definition:Contradiction|contradicts]] the result [[Partition Topology is not T0|Partition Topology is not $T_0$]].
Hence the result, by [[P... | Partition Space is not T1 | https://proofwiki.org/wiki/Partition_Space_is_not_T1 | https://proofwiki.org/wiki/Partition_Space_is_not_T1 | [
"Partition Topologies",
"Examples of T1 Spaces"
] | [
"Definition:Set",
"Definition:Set Partition",
"Definition:Trivial Partition/Partition of Singletons",
"Definition:Partition Topology",
"Basis for Partition Topology",
"Definition:T1 Space"
] | [
"Definition:T1 Space",
"T1 Space is T0",
"Definition:T0 Space",
"Definition:Contradiction",
"Partition Topology is not T0",
"Proof by Contradiction"
] |
proofwiki-13984 | Odd-Even Topology is Lindelöf | Let $T = \struct {\Z_{>0}, \tau}$ be a topological space where $\tau$ is the odd-even topology on the strictly positive integers $\Z_{>0}$.
Then $T$ is Lindelöf. | From Odd-Even Topology is Second-Countable, $T$ is second-countable.
The result follows from Second-Countable Space is Lindelöf.
{{qed}} | Let $T = \struct {\Z_{>0}, \tau}$ be a [[Definition:Topological Space|topological space]] where $\tau$ is the [[Definition:Odd-Even Topology|odd-even topology]] on the [[Definition:Strictly Positive Integer|strictly positive integers]] $\Z_{>0}$.
Then $T$ is [[Definition:Lindelöf Space|Lindelöf]]. | From [[Odd-Even Topology is Second-Countable]], $T$ is [[Definition:Second-Countable Space|second-countable]].
The result follows from [[Second-Countable Space is Lindelöf]].
{{qed}} | Odd-Even Topology is Lindelöf | https://proofwiki.org/wiki/Odd-Even_Topology_is_Lindelöf | https://proofwiki.org/wiki/Odd-Even_Topology_is_Lindelöf | [
"Odd-Even Topology",
"Examples of Lindelöf Spaces"
] | [
"Definition:Topological Space",
"Definition:Odd-Even Topology",
"Definition:Strictly Positive/Integer",
"Definition:Lindelöf Space"
] | [
"Odd-Even Topology is Second-Countable",
"Definition:Second-Countable Space",
"Second-Countable Space is Lindelöf"
] |
proofwiki-13985 | Odd-Even Topology is Separable | Let $T = \struct {\Z_{>0}, \tau}$ be a topological space where $\tau$ is the odd-even topology on the strictly positive integers $\Z_{>0}$.
Then $T$ is separable. | From Odd-Even Topology is Second-Countable, $T$ is second-countable.
The result follows from Second-Countable Space is Separable.
{{qed}} | Let $T = \struct {\Z_{>0}, \tau}$ be a [[Definition:Topological Space|topological space]] where $\tau$ is the [[Definition:Odd-Even Topology|odd-even topology]] on the [[Definition:Strictly Positive Integer|strictly positive integers]] $\Z_{>0}$.
Then $T$ is [[Definition:Separable Space|separable]]. | From [[Odd-Even Topology is Second-Countable]], $T$ is [[Definition:Second-Countable Space|second-countable]].
The result follows from [[Second-Countable Space is Separable]].
{{qed}} | Odd-Even Topology is Separable | https://proofwiki.org/wiki/Odd-Even_Topology_is_Separable | https://proofwiki.org/wiki/Odd-Even_Topology_is_Separable | [
"Odd-Even Topology",
"Examples of Separable Spaces"
] | [
"Definition:Topological Space",
"Definition:Odd-Even Topology",
"Definition:Strictly Positive/Integer",
"Definition:Separable Space"
] | [
"Odd-Even Topology is Second-Countable",
"Definition:Second-Countable Space",
"Second-Countable Space is Separable"
] |
proofwiki-13986 | Odd-Even Topology is First-Countable | Let $T = \struct {\Z_{>0}, \tau}$ be a topological space where $\tau$ is the odd-even topology on the strictly positive integers $\Z_{>0}$.
Then $T$ is first-countable. | From Odd-Even Topology is Second-Countable, $T$ is second-countable.
The result follows from Second-Countable Space is First-Countable Space.
{{qed}} | Let $T = \struct {\Z_{>0}, \tau}$ be a [[Definition:Topological Space|topological space]] where $\tau$ is the [[Definition:Odd-Even Topology|odd-even topology]] on the [[Definition:Strictly Positive Integer|strictly positive integers]] $\Z_{>0}$.
Then $T$ is [[Definition:First-Countable Space|first-countable]]. | From [[Odd-Even Topology is Second-Countable]], $T$ is [[Definition:Second-Countable Space|second-countable]].
The result follows from [[Second-Countable Space is First-Countable Space]].
{{qed}} | Odd-Even Topology is First-Countable | https://proofwiki.org/wiki/Odd-Even_Topology_is_First-Countable | https://proofwiki.org/wiki/Odd-Even_Topology_is_First-Countable | [
"Odd-Even Topology",
"Examples of First-Countable Spaces"
] | [
"Definition:Topological Space",
"Definition:Odd-Even Topology",
"Definition:Strictly Positive/Integer",
"Definition:First-Countable Space"
] | [
"Odd-Even Topology is Second-Countable",
"Definition:Second-Countable Space",
"Second-Countable Space is First-Countable"
] |
proofwiki-13987 | Particular Point Space less Particular Point is Discrete | Let $T = \struct {S, \tau_p}$ be a particular point space, whose particular point is $p$.
Let $H = S \setminus \set p$ where $\setminus$ denotes set difference.
Then the topological subspace $T_H = \struct {H, \tau_H}$ induced on $H$ by $\tau_p$ is a discrete space. | Let $H = S \setminus \set p$.
Let $V \subseteq H$ be any subset of $H$.
As $p \notin V$, $V$ is a closed set of $T$.
Thus, by definition of closed set $S \setminus V$ is open in $T$.
By definition of subspace topology, $\paren {S \setminus V} \cap H$ is open in $T_H$.
From Intersection with Set Difference is Set Differ... | Let $T = \struct {S, \tau_p}$ be a [[Definition:Particular Point Space|particular point space]], whose [[Definition:Particular Point|particular point]] is $p$.
Let $H = S \setminus \set p$ where $\setminus$ denotes [[Definition:Set Difference|set difference]].
Then the [[Definition:Topological Subspace|topological s... | Let $H = S \setminus \set p$.
Let $V \subseteq H$ be any [[Definition:Subset|subset]] of $H$.
As $p \notin V$, $V$ is a [[Definition:Closed Set (Topology)|closed set]] of $T$.
Thus, by definition of [[Definition:Closed Set (Topology)|closed set]] $S \setminus V$ is [[Definition:Open Set (Topology)|open]] in $T$.
B... | Particular Point Space less Particular Point is Discrete | https://proofwiki.org/wiki/Particular_Point_Space_less_Particular_Point_is_Discrete | https://proofwiki.org/wiki/Particular_Point_Space_less_Particular_Point_is_Discrete | [
"Particular Point Topologies",
"Discrete Topologies"
] | [
"Definition:Particular Point Topology",
"Definition:Particular Point",
"Definition:Set Difference",
"Definition:Topological Subspace",
"Definition:Discrete Topology"
] | [
"Definition:Subset",
"Definition:Closed Set/Topology",
"Definition:Closed Set/Topology",
"Definition:Open Set/Topology",
"Definition:Topological Subspace",
"Definition:Open Set/Topology",
"Intersection with Set Difference is Set Difference with Intersection",
"Intersection with Subset is Subset",
"D... |
proofwiki-13988 | Zero is Accumulation Point of Sequence in Sierpiński Space | Let $T = \struct {\set {0, 1}, \tau_0}$ be a Sierpiński space.
The sequence in $T$:
:$\sigma = \sequence {0, 1, 0, 1, \ldots}$
has $0$ as an accumulation point. | By definition, $\alpha$ is an accumulation point of $\sigma$ {{iff}}:
:$\forall U \in \tau_0: \alpha \in U \implies \set {n \in \N: x_n \in U}$ is infinite.
Both $\set 0$ and $\set {0, 1}$ contain $0$, which occurs an infinite number of times in $\sigma$.
Hence, by definition, $0$ is an accumulation point of $\sigma$.
... | Let $T = \struct {\set {0, 1}, \tau_0}$ be a [[Definition:Sierpiński Space|Sierpiński space]].
The [[Definition:Sequence|sequence]] in $T$:
:$\sigma = \sequence {0, 1, 0, 1, \ldots}$
has $0$ as an [[Definition:Accumulation Point of Sequence|accumulation point]]. | By definition, $\alpha$ is an [[Definition:Accumulation Point of Sequence|accumulation point]] of $\sigma$ {{iff}}:
:$\forall U \in \tau_0: \alpha \in U \implies \set {n \in \N: x_n \in U}$ is [[Definition:Infinite Set|infinite]].
Both $\set 0$ and $\set {0, 1}$ contain $0$, which occurs an [[Definition:Infinite Set|i... | Zero is Accumulation Point of Sequence in Sierpiński Space | https://proofwiki.org/wiki/Zero_is_Accumulation_Point_of_Sequence_in_Sierpiński_Space | https://proofwiki.org/wiki/Zero_is_Accumulation_Point_of_Sequence_in_Sierpiński_Space | [
"Sierpiński Space",
"Examples of Accumulation Points"
] | [
"Definition:Sierpiński Space",
"Definition:Sequence",
"Definition:Accumulation Point/Sequence"
] | [
"Definition:Accumulation Point/Sequence",
"Definition:Infinite Set",
"Definition:Infinite Set",
"Definition:Accumulation Point/Sequence"
] |
proofwiki-13989 | 1 is Limit Point of Sequence in Sierpiński Space | Let $T = \struct {\set {0, 1}, \tau_0}$ be a Sierpiński space.
The sequence in $T$:
:$\sigma = \sequence {0, 1, 0, 1, \ldots}$
has $1$ as a limit. | By definition, $\alpha$ is a limit of $\sigma$ {{iff}}:
:$\forall U \in \tau_0: \alpha \in U \implies \set {n \in \N: x_n \notin U}$ is finite.
The only open set of $T$ containing $1$ is $\set {0, 1}$.
It contains all but a finite number (that is: $0$) elements of $\sigma$.
Hence, by definition, $1$ is a limit of $\sig... | Let $T = \struct {\set {0, 1}, \tau_0}$ be a [[Definition:Sierpiński Space|Sierpiński space]].
The [[Definition:Sequence|sequence]] in $T$:
:$\sigma = \sequence {0, 1, 0, 1, \ldots}$
has $1$ as a [[Definition:Limit of Sequence (Topology)|limit]]. | By definition, $\alpha$ is a [[Definition:Limit of Sequence (Topology)|limit]] of $\sigma$ {{iff}}:
:$\forall U \in \tau_0: \alpha \in U \implies \set {n \in \N: x_n \notin U}$ is [[Definition:Finite Set|finite]].
The only [[Definition:Open Set (Topology)|open set]] of $T$ containing $1$ is $\set {0, 1}$.
It contains... | 1 is Limit Point of Sequence in Sierpiński Space | https://proofwiki.org/wiki/1_is_Limit_Point_of_Sequence_in_Sierpiński_Space | https://proofwiki.org/wiki/1_is_Limit_Point_of_Sequence_in_Sierpiński_Space | [
"Sierpiński Space",
"Examples of Limit Points"
] | [
"Definition:Sierpiński Space",
"Definition:Sequence",
"Definition:Limit of Sequence/Topological Space"
] | [
"Definition:Limit of Sequence/Topological Space",
"Definition:Finite Set",
"Definition:Open Set/Topology",
"Definition:Finite Set",
"Definition:Limit of Sequence/Topological Space"
] |
proofwiki-13990 | Infinite Particular Point Space is not Metacompact | Let $T = \struct {S, \tau_p}$ be an infinite particular point space.
Then $T$ is not metacompact. | {{AimForCont}} $T$ is metacompact.
From Metacompact Space is Countably Metacompact it follows that $T$ is countably metacompact.
But we have that Infinite Particular Point Space is not Countably Metacompact.
Hence the result by Proof by Contradiction.
{{qed}}
Category:Particular Point Topologies
Category:Examples of Me... | Let $T = \struct {S, \tau_p}$ be an [[Definition:Infinite Particular Point Topology|infinite particular point space]].
Then $T$ is not [[Definition:Metacompact Space|metacompact]]. | {{AimForCont}} $T$ is [[Definition:Metacompact Space|metacompact]].
From [[Metacompact Space is Countably Metacompact]] it follows that $T$ is [[Definition:Countably Metacompact Space|countably metacompact]].
But we have that [[Infinite Particular Point Space is not Countably Metacompact]].
Hence the result by [[Pro... | Infinite Particular Point Space is not Metacompact | https://proofwiki.org/wiki/Infinite_Particular_Point_Space_is_not_Metacompact | https://proofwiki.org/wiki/Infinite_Particular_Point_Space_is_not_Metacompact | [
"Particular Point Topologies",
"Examples of Metacompact Spaces"
] | [
"Definition:Particular Point Topology/Infinite",
"Definition:Metacompact Space"
] | [
"Definition:Metacompact Space",
"Metacompact Space is Countably Metacompact",
"Definition:Countably Metacompact Space",
"Infinite Particular Point Space is not Countably Metacompact",
"Proof by Contradiction",
"Category:Particular Point Topologies",
"Category:Examples of Metacompact Spaces"
] |
proofwiki-13991 | Infinite Particular Point Space is not Countably Paracompact | Let $T = \struct {S, \tau_p}$ be an infinite particular point space.
Then $T$ is not countably paracompact. | {{AimForCont}} $T$ is countably paracompact.
From Countably Paracompact Space is Countably Metacompact it follows that $T$ is countably metacompact.
But we have that Infinite Particular Point Space is not Countably Metacompact.
Hence the result by Proof by Contradiction.
{{qed}}
Category:Particular Point Topologies
Cat... | Let $T = \struct {S, \tau_p}$ be an [[Definition:Infinite Particular Point Topology|infinite particular point space]].
Then $T$ is not [[Definition:Countably Paracompact Space|countably paracompact]]. | {{AimForCont}} $T$ is [[Definition:Countably Paracompact Space|countably paracompact]].
From [[Countably Paracompact Space is Countably Metacompact]] it follows that $T$ is [[Definition:Countably Metacompact Space|countably metacompact]].
But we have that [[Infinite Particular Point Space is not Countably Metacompact... | Infinite Particular Point Space is not Countably Paracompact | https://proofwiki.org/wiki/Infinite_Particular_Point_Space_is_not_Countably_Paracompact | https://proofwiki.org/wiki/Infinite_Particular_Point_Space_is_not_Countably_Paracompact | [
"Particular Point Topologies",
"Examples of Countably Paracompact Spaces"
] | [
"Definition:Particular Point Topology/Infinite",
"Definition:Countably Paracompact Space"
] | [
"Definition:Countably Paracompact Space",
"Countably Paracompact Space is Countably Metacompact",
"Definition:Countably Metacompact Space",
"Infinite Particular Point Space is not Countably Metacompact",
"Proof by Contradiction",
"Category:Particular Point Topologies",
"Category:Examples of Countably Pa... |
proofwiki-13992 | Closed Extension Topology is not T2 | Let $T = \struct {S, \tau}$ be a topological space.
Let $T^*_p = \struct {S^*_p, \tau^*_p}$ be the closed extension space of $T$.
Then $T^*_p$ is not a $T_2$ (Hausdorff) space. | {{AimForCont}} $T^*_p$ is not a $T_2$ space.
From $T_2$ Space is $T_1$ Space, $T^*_p$ is a $T_1$ space.
But this contradicts Closed Extension Topology is not $T_1$
Hence by Proof by Contradiction $T^*_p$ can not be a $T_2$ space.
{{qed}} | Let $T = \struct {S, \tau}$ be a [[Definition:Topological Space|topological space]].
Let $T^*_p = \struct {S^*_p, \tau^*_p}$ be the [[Definition:Closed Extension Space|closed extension space]] of $T$.
Then $T^*_p$ is not a [[Definition:T2 Space|$T_2$ (Hausdorff) space]]. | {{AimForCont}} $T^*_p$ is not a [[Definition:T2 Space|$T_2$ space]].
From [[T2 Space is T1 Space|$T_2$ Space is $T_1$ Space]], $T^*_p$ is a [[Definition:T1 Space|$T_1$ space]].
But this contradicts [[Closed Extension Topology is not T1|Closed Extension Topology is not $T_1$]]
Hence by [[Proof by Contradiction]] $T^*... | Closed Extension Topology is not T2 | https://proofwiki.org/wiki/Closed_Extension_Topology_is_not_T2 | https://proofwiki.org/wiki/Closed_Extension_Topology_is_not_T2 | [
"Closed Extension Topologies",
"Examples of Hausdorff Spaces"
] | [
"Definition:Topological Space",
"Definition:Closed Extension Topology",
"Definition:T2 Space"
] | [
"Definition:T2 Space",
"T2 Space is T1",
"Definition:T1 Space",
"Closed Extension Topology is not T1",
"Proof by Contradiction",
"Definition:T2 Space"
] |
proofwiki-13993 | Closed Extension Topology is not T3 | Let $T = \struct {S, \tau}$ be a topological space.
Let $T^*_p = \struct {S^*_p, \tau^*_p}$ be the closed extension space of $T$.
Then $T^*_p$ is not a $T_3$ space. | By Underlying Set of Topological Space is Closed, $S$ is closed in $T$.
By Closed Sets of Closed Extension Topology, $S$ is closed in $T^*_p$.
{{Defof|Closed Extension Space}} gives:
:$p \notin S$
:Every open set in $T^*_p$ is either $\O$ or it contains $p$.
Thus no open set containing $S$ is disjoint from $\set p$.
Th... | Let $T = \struct {S, \tau}$ be a [[Definition:Topological Space|topological space]].
Let $T^*_p = \struct {S^*_p, \tau^*_p}$ be the [[Definition:Closed Extension Space|closed extension space]] of $T$.
Then $T^*_p$ is not a [[Definition:T3 Space|$T_3$ space]]. | By [[Underlying Set of Topological Space is Closed]], $S$ is [[Definition:Closed Set (Topology)|closed]] in $T$.
By [[Closed Sets of Closed Extension Topology]], $S$ is [[Definition:Closed Set (Topology)|closed]] in $T^*_p$.
{{Defof|Closed Extension Space}} gives:
:$p \notin S$
:Every [[Definition:Open Set (Topology)... | Closed Extension Topology is not T3 | https://proofwiki.org/wiki/Closed_Extension_Topology_is_not_T3 | https://proofwiki.org/wiki/Closed_Extension_Topology_is_not_T3 | [
"Closed Extension Topologies",
"Examples of T3 Spaces"
] | [
"Definition:Topological Space",
"Definition:Closed Extension Topology",
"Definition:T3 Space"
] | [
"Underlying Set of Topological Space is Closed",
"Definition:Closed Set/Topology",
"Closed Sets of Closed Extension Topology",
"Definition:Closed Set/Topology",
"Definition:Open Set/Topology",
"Definition:Open Set/Topology",
"Definition:Disjoint Sets",
"Definition:T3 Space"
] |
proofwiki-13994 | Condition for Closed Extension Space to be T4 Space | Let $T = \struct {S, \tau}$ be a topological space.
Let $T^*_p = \struct {S^*_p, \tau^*_p}$ be the closed extension space of $T$.
Then:
:$T^*_p$ is a $T_4$ space {{iff}} $T$ is a $T_4$ space vacuously
and $T^*_p$ in this case is also a $T_4$ space vacuously. | === Sufficient Condition ===
Suppose $T^*_p$ is $T_4$.
Then for any two disjoint closed sets $A, B \subseteq S$ there exist disjoint open sets $U, V \in \tau^*_p$ containing $A$ and $B$ respectively.
However, for any non-empty set $U \in \tau^*_p$, $p \in U$.
Hence no non-empty open sets in $T^*_p$ are disjoint.
Theref... | Let $T = \struct {S, \tau}$ be a [[Definition:Topological Space|topological space]].
Let $T^*_p = \struct {S^*_p, \tau^*_p}$ be the [[Definition:Closed Extension Space|closed extension space]] of $T$.
Then:
:$T^*_p$ is a [[Definition:T4 Space|$T_4$ space]] {{iff}} $T$ is a [[Definition:T4 Space|$T_4$ space]] [[Defin... | === Sufficient Condition ===
Suppose $T^*_p$ is [[Definition:T4 Space|$T_4$]].
Then for any two [[Definition:Disjoint Sets|disjoint]] [[Definition:Closed Set (Topology)|closed sets]] $A, B \subseteq S$ there exist [[Definition:Disjoint Sets|disjoint]] [[Definition:Open Set (Topology)|open sets]] $U, V \in \tau^*_p$ c... | Condition for Closed Extension Space to be T4 Space | https://proofwiki.org/wiki/Condition_for_Closed_Extension_Space_to_be_T4_Space | https://proofwiki.org/wiki/Condition_for_Closed_Extension_Space_to_be_T4_Space | [
"Closed Extension Topologies",
"Examples of T4 Spaces"
] | [
"Definition:Topological Space",
"Definition:Closed Extension Topology",
"Definition:T4 Space",
"Definition:T4 Space",
"Definition:Vacuous Truth",
"Definition:T4 Space",
"Definition:Vacuous Truth"
] | [
"Definition:T4 Space",
"Definition:Disjoint Sets",
"Definition:Closed Set/Topology",
"Definition:Disjoint Sets",
"Definition:Open Set/Topology",
"Definition:Non-Empty Set",
"Definition:Non-Empty Set",
"Definition:Open Set/Topology",
"Definition:Disjoint Sets",
"Definition:T4 Space",
"Definition:... |
proofwiki-13995 | Condition for Closed Extension Space to be T5 Space | Let $T = \struct {S, \tau}$ be a topological space.
Let $T^*_p = \struct {S^*_p, \tau^*_p}$ be the closed extension space of $T$.
Then:
:$T^*_p$ is a $T_5$ space {{iff}} $T$ is a $T_5$ space vacuously
and $T^*_p$ in this case is also a $T_5$ space vacuously. | === Sufficient Condition ===
Let $T^*_p$ be a $T_5$ space.
Then for any two separated sets $A, B \subseteq S$ there exist disjoint open sets $U, V \in \tau^*_p$ containing $A$ and $B$ respectively.
However, for any non-empty set $U \in \tau^*_p$:
:$p \in U$
Hence no non-empty open sets in $T^*_p$ are separated.
Therefo... | Let $T = \struct {S, \tau}$ be a [[Definition:Topological Space|topological space]].
Let $T^*_p = \struct {S^*_p, \tau^*_p}$ be the [[Definition:Closed Extension Space|closed extension space]] of $T$.
Then:
:$T^*_p$ is a [[Definition:T5 Space|$T_5$ space]] {{iff}} $T$ is a [[Definition:T5 Space|$T_5$ space]] [[Defin... | === Sufficient Condition ===
Let $T^*_p$ be a [[Definition:T5 Space|$T_5$ space]].
Then for any two [[Definition:Separated Sets|separated sets]] $A, B \subseteq S$ there exist [[Definition:Disjoint Sets|disjoint]] [[Definition:Open Set (Topology)|open sets]] $U, V \in \tau^*_p$ containing $A$ and $B$ respectively.
H... | Condition for Closed Extension Space to be T5 Space | https://proofwiki.org/wiki/Condition_for_Closed_Extension_Space_to_be_T5_Space | https://proofwiki.org/wiki/Condition_for_Closed_Extension_Space_to_be_T5_Space | [
"Closed Extension Topologies",
"Examples of T5 Spaces"
] | [
"Definition:Topological Space",
"Definition:Closed Extension Topology",
"Definition:T5 Space",
"Definition:T5 Space",
"Definition:Vacuous Truth",
"Definition:T5 Space",
"Definition:Vacuous Truth"
] | [
"Definition:T5 Space",
"Definition:Separated Sets",
"Definition:Disjoint Sets",
"Definition:Open Set/Topology",
"Definition:Non-Empty Set",
"Definition:Non-Empty Set",
"Definition:Open Set/Topology",
"Definition:Separated Sets",
"Definition:T5 Space",
"Definition:Vacuous Truth",
"Definition:Sepa... |
proofwiki-13996 | Either-Or Topology is T4 | Let $T = \struct {S, \tau}$ be the either-or space.
Then $T$ is a $T_4$ space. | Follows from:
:Either-Or Topology is $T_5$
:$T_5$ Space is $T_4$.
{{qed}} | Let $T = \struct {S, \tau}$ be the [[Definition:Either-Or Space|either-or space]].
Then $T$ is a [[Definition:T4 Space|$T_4$ space]]. | Follows from:
:[[Either-Or Topology is T5|Either-Or Topology is $T_5$]]
:[[T5 Space is T4|$T_5$ Space is $T_4$]].
{{qed}} | Either-Or Topology is T4 | https://proofwiki.org/wiki/Either-Or_Topology_is_T4 | https://proofwiki.org/wiki/Either-Or_Topology_is_T4 | [
"Either-Or Topology",
"Examples of T4 Spaces"
] | [
"Definition:Either-Or Topology",
"Definition:T4 Space"
] | [
"Either-Or Topology is T5",
"T5 Space is T4"
] |
proofwiki-13997 | Limit Points in Fort Space | Let $T = \struct {S, \tau_p}$ be a Fort space.
Let $x \in S$ such that $x \ne p$.
Then $p$ is the only limit point of $x$. | From {{Defof|Fort Space}}, we have $\relcomp S {\set x} \in \tau_p$.
For any $y \ne x$, $y \in \relcomp S {\set x}$.
Therefore $\relcomp s {\set x}$ is an open neighborhood of $y$.
From {{Defof|Relative Complement}} we also have $x \notin \relcomp S {\set x}$.
Hence $y$ is not a limit point of $x$.
By {{Defof|Limit Poi... | Let $T = \struct {S, \tau_p}$ be a [[Definition:Fort Space|Fort space]].
Let $x \in S$ such that $x \ne p$.
Then $p$ is the only [[Definition:Limit Point of Point|limit point]] of $x$. | From {{Defof|Fort Space}}, we have $\relcomp S {\set x} \in \tau_p$.
For any $y \ne x$, $y \in \relcomp S {\set x}$.
Therefore $\relcomp s {\set x}$ is an [[Definition:Open Neighborhood of Point|open neighborhood]] of $y$.
From {{Defof|Relative Complement}} we also have $x \notin \relcomp S {\set x}$.
Hence $y$ is ... | Limit Points in Fort Space | https://proofwiki.org/wiki/Limit_Points_in_Fort_Space | https://proofwiki.org/wiki/Limit_Points_in_Fort_Space | [
"Fort Spaces",
"Examples of Limit Points"
] | [
"Definition:Fort Space",
"Definition:Limit Point/Topology/Point"
] | [
"Definition:Open Neighborhood/Point",
"Definition:Limit Point/Topology/Point",
"Definition:Limit Point/Topology/Point",
"Definition:Limit Point/Topology/Point"
] |
proofwiki-13998 | Real Number Line is First-Countable | Let $\struct {\R, \tau_d}$ be the real number line with the usual (Euclidean) topology.
Then $\struct {\R, \tau_d}$ is a first-countable space. | From Real Number Line is Second-Countable we have that $\struct {\R, \tau_d}$ is a second-countable space.
The result follows from Second-Countable Space is First-Countable.
{{qed}} | Let $\struct {\R, \tau_d}$ be the [[Definition:Real Number Line with Euclidean Topology|real number line with the usual (Euclidean) topology]].
Then $\struct {\R, \tau_d}$ is a [[Definition:First-Countable Space|first-countable space]]. | From [[Real Number Line is Second-Countable]] we have that $\struct {\R, \tau_d}$ is a [[Definition:Second-Countable Space|second-countable space]].
The result follows from [[Second-Countable Space is First-Countable]].
{{qed}} | Real Number Line is First-Countable | https://proofwiki.org/wiki/Real_Number_Line_is_First-Countable | https://proofwiki.org/wiki/Real_Number_Line_is_First-Countable | [
"Real Number Line with Euclidean Topology",
"Examples of First-Countable Spaces"
] | [
"Definition:Euclidean Space/Euclidean Topology/Real Number Line",
"Definition:First-Countable Space"
] | [
"Real Number Line is Second-Countable",
"Definition:Second-Countable Space",
"Second-Countable Space is First-Countable"
] |
proofwiki-13999 | Alexandroff Extension is Compact | Let $T = \struct {S, \tau}$ be a non-empty topological space.
Let $p$ be a new element not in $S$.
Let $S^* := S \cup \set p$.
Let $T^* = \struct {S^*, \tau^*}$ be the Alexandroff extension on $S$.
Then $T^*$ is a compact topological space. | Let $\UU$ be an open cover of $T^*$.
At least one $V \in \UU$ contains $p$.
Because $p \notin S$, $V$ is not an open set of $T$.
Therefore, by definition of the Alexandroff extension, $V$ must be the complement relative to $S^*$ of a closed, compact subset $\relcomp {S^*} V$ of $T$.
Because $\relcomp {S^*} V$ is compac... | Let $T = \struct {S, \tau}$ be a [[Definition:Non-Empty Set|non-empty]] [[Definition:Topological Space|topological space]].
Let $p$ be a [[Definition:New Element|new element]] not in $S$.
Let $S^* := S \cup \set p$.
Let $T^* = \struct {S^*, \tau^*}$ be the [[Definition:Alexandroff Extension|Alexandroff extension]] o... | Let $\UU$ be an [[Definition:Open Cover|open cover]] of $T^*$.
At least one $V \in \UU$ contains $p$.
Because $p \notin S$, $V$ is not an [[Definition:Open Set (Topology)|open set]] of $T$.
Therefore, by definition of the [[Definition:Alexandroff Extension|Alexandroff extension]], $V$ must be the [[Definition:Relati... | Alexandroff Extension is Compact | https://proofwiki.org/wiki/Alexandroff_Extension_is_Compact | https://proofwiki.org/wiki/Alexandroff_Extension_is_Compact | [
"Alexandroff Extensions",
"Examples of Compact Topological Spaces"
] | [
"Definition:Non-Empty Set",
"Definition:Topological Space",
"Definition:New Element",
"Definition:Alexandroff Extension",
"Definition:Compact Topological Space"
] | [
"Definition:Open Cover",
"Definition:Open Set/Topology",
"Definition:Alexandroff Extension",
"Definition:Relative Complement",
"Definition:Closed Set/Topology",
"Definition:Compact Topological Space/Subspace",
"Definition:Subset",
"Definition:Compact Topological Space/Subspace",
"Definition:Cover of... |
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