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-3800 | Condition for Open Extension Space to be T0 | Let $T = \struct {S, \tau}$ be a topological space.
Let $T^*_{\bar p} = \struct {S^*_p, \tau^*_{\bar p} }$ be the open extension space of $T$.
Then $T^*_{\bar p}$ is a $T_0$ space {{iff}} $T$ is. | By definition:
:$\tau^*_{\bar p} = \set {U: U \in \tau} \cup \set {S^*_p}$
Let $T = \struct {S, \tau}$ be a $T_0$ space.
Let $\forall x, y \in S$ such that $x \ne y$.
First, suppose $x \ne p \ne y$.
Then by definition of $T$ as a $T_0$ space:
:$\exists U \in \tau: x \in U, y \notin U$
or:
:$\exists U \in \tau: y \in U,... | Let $T = \struct {S, \tau}$ be a [[Definition:Topological Space|topological space]].
Let $T^*_{\bar p} = \struct {S^*_p, \tau^*_{\bar p} }$ be the [[Definition:Open Extension Space|open extension space]] of $T$.
Then $T^*_{\bar p}$ is a [[Definition:T0 Space|$T_0$ space]] {{iff}} $T$ is. | By definition:
:$\tau^*_{\bar p} = \set {U: U \in \tau} \cup \set {S^*_p}$
Let $T = \struct {S, \tau}$ be a [[Definition:T0 Space|$T_0$ space]].
Let $\forall x, y \in S$ such that $x \ne y$.
First, suppose $x \ne p \ne y$.
Then by definition of $T$ as a [[Definition:T0 Space|$T_0$ space]]:
:$\exists U \in \tau: x... | Condition for Open Extension Space to be T0 | https://proofwiki.org/wiki/Condition_for_Open_Extension_Space_to_be_T0 | https://proofwiki.org/wiki/Condition_for_Open_Extension_Space_to_be_T0 | [
"Open Extension Topologies",
"Examples of T0 Spaces"
] | [
"Definition:Topological Space",
"Definition:Open Extension Topology",
"Definition:T0 Space"
] | [
"Definition:T0 Space",
"Definition:T0 Space",
"Definition:T0 Space",
"Definition:T0 Space",
"Definition:T0 Space"
] |
proofwiki-3801 | Non-Trivial Excluded Point Topology is not T1 | Let $T = \struct {S, \tau_{\bar p} }$ be a excluded point space such that $S$ is not a singleton.
Then $T$ is not a $T_1$ space. | Follows directly from:
:Excluded Point Topology is Open Extension Topology of Discrete Topology
:Open Extension Topology is not $T_1$
{{qed}} | Let $T = \struct {S, \tau_{\bar p} }$ be a [[Definition:Excluded Point Topology|excluded point space]] such that $S$ is not a [[Definition:Singleton|singleton]].
Then $T$ is not a [[Definition:T1 Space|$T_1$ space]]. | Follows directly from:
:[[Excluded Point Topology is Open Extension Topology of Discrete Topology]]
:[[Open Extension Topology is not T1|Open Extension Topology is not $T_1$]]
{{qed}} | Non-Trivial Excluded Point Topology is not T1 | https://proofwiki.org/wiki/Non-Trivial_Excluded_Point_Topology_is_not_T1 | https://proofwiki.org/wiki/Non-Trivial_Excluded_Point_Topology_is_not_T1 | [
"Excluded Point Topologies",
"Examples of T1 Spaces"
] | [
"Definition:Excluded Point Topology",
"Definition:Singleton",
"Definition:T1 Space"
] | [
"Excluded Point Topology is Open Extension Topology of Discrete Topology",
"Open Extension Topology is not T1"
] |
proofwiki-3802 | Limit Points in Open Extension Space | Let $T = \struct {S, \tau}$ be a topological space.
Let $T^*_{\bar p} = \struct {S^*_p, \tau^*_{\bar p} }$ be the open extension space of $T$.
Let $x \in S$.
Then $p$ is a limit point of $x$. | Every open set of $T^*_p = \struct {S^*_p, \tau^*_{\bar p} }$ except $S^*_p$ does not contain the point $p$ by definition.
So every open set $U \in \tau^*_{\bar p}$ such that $p \in U$ (there is only the one such open set) contains $x$.
So by definition of the limit point of a point, $p$ is a limit point of $x$.
{{qed}... | Let $T = \struct {S, \tau}$ be a [[Definition:Topological Space|topological space]].
Let $T^*_{\bar p} = \struct {S^*_p, \tau^*_{\bar p} }$ be the [[Definition:Open Extension Space|open extension space]] of $T$.
Let $x \in S$.
Then $p$ is a [[Definition:Limit Point of Point|limit point]] of $x$. | Every [[Definition:Open Set (Topology)|open set]] of $T^*_p = \struct {S^*_p, \tau^*_{\bar p} }$ except $S^*_p$ does not contain the point $p$ by [[Definition:Open Extension Topology|definition]].
So every [[Definition:Open Set (Topology)|open set]] $U \in \tau^*_{\bar p}$ such that $p \in U$ (there is only the one su... | Limit Points in Open Extension Space | https://proofwiki.org/wiki/Limit_Points_in_Open_Extension_Space | https://proofwiki.org/wiki/Limit_Points_in_Open_Extension_Space | [
"Limit Points in Open Extension Space",
"Open Extension Topologies",
"Examples of Limit Points"
] | [
"Definition:Topological Space",
"Definition:Open Extension Topology",
"Definition:Limit Point/Topology/Point"
] | [
"Definition:Open Set/Topology",
"Definition:Open Extension Topology",
"Definition:Open Set/Topology",
"Definition:Open Set/Topology",
"Definition:Limit Point/Topology/Point",
"Definition:Limit Point/Topology/Point",
"Category:Limit Points in Open Extension Space",
"Category:Open Extension Topologies",... |
proofwiki-3803 | Limit Points in Excluded Point Space | Let $T = \struct {S, \tau_{\bar p} }$ be an excluded point space.
Let $x \in S$ such that $x \ne p$.
Then $p$ is the only limit point of $x$.
Similarly, let $U \subseteq S$.
Then $p$ is the only limit point of $U$. | Let $U \subseteq S$.
Let $x \in S$ such that $x \ne p$.
From:
:Excluded Point Topology is Open Extension Topology of Discrete Topology
:Limit Points in Open Extension Space
it follows that:
:$p$ is a limit point of $U$
:$p$ is a limit point of $x$.
Now suppose $y \in S$ such that $y \ne p$.
We have by definition of exc... | Let $T = \struct {S, \tau_{\bar p} }$ be an [[Definition:Excluded Point Space|excluded point space]].
Let $x \in S$ such that $x \ne p$.
Then $p$ is the only [[Definition:Limit Point of Point|limit point]] of $x$.
Similarly, let $U \subseteq S$.
Then $p$ is the only [[Definition:Limit Point of Set|limit point]] of... | Let $U \subseteq S$.
Let $x \in S$ such that $x \ne p$.
From:
:[[Excluded Point Topology is Open Extension Topology of Discrete Topology]]
:[[Limit Points in Open Extension Space]]
it follows that:
:$p$ is a [[Definition:Limit Point of Set|limit point of $U$]]
:$p$ is a [[Definition:Limit Point of Point|limit point ... | Limit Points in Excluded Point Space | https://proofwiki.org/wiki/Limit_Points_in_Excluded_Point_Space | https://proofwiki.org/wiki/Limit_Points_in_Excluded_Point_Space | [
"Excluded Point Topologies",
"Examples of Limit Points"
] | [
"Definition:Excluded Point Topology",
"Definition:Limit Point/Topology/Point",
"Definition:Limit Point/Topology/Set"
] | [
"Excluded Point Topology is Open Extension Topology of Discrete Topology",
"Limit Points in Open Extension Space",
"Definition:Limit Point/Topology/Set",
"Definition:Limit Point/Topology/Point",
"Definition:Excluded Point Topology",
"Definition:Open Set/Topology",
"Definition:Limit Point/Topology/Set",
... |
proofwiki-3804 | Excluded Point Space is T5 | Let $T = \struct {S, \tau_{\bar p} }$ be an excluded point space.
Then $T$ is a $T_5$ space. | Let $A, B \subseteq S$ such that:
:$A^- \cap B = A \cap B^- = \O$
where $A^-$ denotes the closure of $A$.
Every closed set of $T$ contains $\set p$.
From Limit Points in Excluded Point Space and the definition of closure:
:$A^- = A \cup \set p, B^- = B \cup \set p$
So if $A^- \cap B = A \cap B^- = \O$ then $p \in A^-$ ... | Let $T = \struct {S, \tau_{\bar p} }$ be an [[Definition:Excluded Point Topology|excluded point space]].
Then $T$ is a [[Definition:T5 Space|$T_5$ space]]. | Let $A, B \subseteq S$ such that:
:$A^- \cap B = A \cap B^- = \O$
where $A^-$ denotes the [[Definition:Closure (Topology)|closure]] of $A$.
Every [[Definition:Closed Set (Topology)|closed set]] of $T$ contains $\set p$.
From [[Limit Points in Excluded Point Space]] and the definition of [[Definition:Closure (Topology... | Excluded Point Space is T5 | https://proofwiki.org/wiki/Excluded_Point_Space_is_T5 | https://proofwiki.org/wiki/Excluded_Point_Space_is_T5 | [
"Excluded Point Topologies",
"Examples of T5 Spaces"
] | [
"Definition:Excluded Point Topology",
"Definition:T5 Space"
] | [
"Definition:Closure (Topology)",
"Definition:Closed Set/Topology",
"Limit Points in Excluded Point Space",
"Definition:Closure (Topology)",
"Definition:Open Set/Topology"
] |
proofwiki-3805 | Excluded Point Space is Compact | Let $T = \struct {S, \tau_{\bar p} }$ be an excluded point space.
Then $T$ is a compact space. | We have:
:Excluded Point Topology is Open Extension Topology of Discrete Topology
:Open Extension Space is Compact
{{qed}} | Let $T = \struct {S, \tau_{\bar p} }$ be an [[Definition:Excluded Point Topology|excluded point space]].
Then $T$ is a [[Definition:Compact Topological Space|compact space]]. | We have:
:[[Excluded Point Topology is Open Extension Topology of Discrete Topology]]
:[[Open Extension Space is Compact]]
{{qed}} | Excluded Point Space is Compact/Proof 1 | https://proofwiki.org/wiki/Excluded_Point_Space_is_Compact | https://proofwiki.org/wiki/Excluded_Point_Space_is_Compact/Proof_1 | [
"Excluded Point Space is Compact",
"Excluded Point Topologies",
"Examples of Compact Topological Spaces"
] | [
"Definition:Excluded Point Topology",
"Definition:Compact Topological Space"
] | [
"Excluded Point Topology is Open Extension Topology of Discrete Topology",
"Open Extension Space is Compact"
] |
proofwiki-3806 | Excluded Point Space is Compact | Let $T = \struct {S, \tau_{\bar p} }$ be an excluded point space.
Then $T$ is a compact space. | By definition of excluded point space, the only open set of $T$ which contains $p$ is $S$.
So any open cover $\CC$ of $T$ must have $S$ in it.
So $\set S$ will be a subcover of $\CC$, whatever $\CC$ may be.
And $\set S$ (having only one set in it) is trivially a finite cover of $T$.
Hence the result, by definition of c... | Let $T = \struct {S, \tau_{\bar p} }$ be an [[Definition:Excluded Point Topology|excluded point space]].
Then $T$ is a [[Definition:Compact Topological Space|compact space]]. | By definition of [[Definition:Excluded Point Space|excluded point space]], the only [[Definition:Open Set (Topology)|open set]] of $T$ which contains $p$ is $S$.
So any [[Definition:Open Cover|open cover]] $\CC$ of $T$ must have $S$ in it.
So $\set S$ will be a [[Definition:Subcover|subcover]] of $\CC$, whatever $\CC... | Excluded Point Space is Compact/Proof 2 | https://proofwiki.org/wiki/Excluded_Point_Space_is_Compact | https://proofwiki.org/wiki/Excluded_Point_Space_is_Compact/Proof_2 | [
"Excluded Point Space is Compact",
"Excluded Point Topologies",
"Examples of Compact Topological Spaces"
] | [
"Definition:Excluded Point Topology",
"Definition:Compact Topological Space"
] | [
"Definition:Excluded Point Topology",
"Definition:Open Set/Topology",
"Definition:Open Cover",
"Definition:Subcover",
"Definition:Cover of Set/Finite",
"Definition:Compact Topological Space"
] |
proofwiki-3807 | Open Extension Space is Compact | Let $T = \struct {S, \tau}$ be a topological space.
Let $T^*_{\bar p} = \struct {S^*_p, \tau^*_{\bar p} }$ be the open extension space of $T$.
Then $T^*_{\bar p}$ is a compact space. | By definition of open extension space, the only open set of $T^*_{\bar p}$ containing $p$ is $S^*_p$.
So any open cover $\CC$ of $T^*_{\bar p}$ must have $S^*_p$ in it.
So $\set {S^*_p}$ will be a subcover of $\CC$, whatever $\CC$ may be.
And $\set {S^*_p}$ (having only one set in it) is trivially a finite cover of $T^... | Let $T = \struct {S, \tau}$ be a [[Definition:Topological Space|topological space]].
Let $T^*_{\bar p} = \struct {S^*_p, \tau^*_{\bar p} }$ be the [[Definition:Open Extension Space|open extension space]] of $T$.
Then $T^*_{\bar p}$ is a [[Definition:Compact Topological Space|compact space]]. | By definition of [[Definition:Open Extension Space|open extension space]], the only [[Definition:Open Set (Topology)|open set]] of $T^*_{\bar p}$ containing $p$ is $S^*_p$.
So any [[Definition:Open Cover|open cover]] $\CC$ of $T^*_{\bar p}$ must have $S^*_p$ in it.
So $\set {S^*_p}$ will be a [[Definition:Subcover|su... | Open Extension Space is Compact | https://proofwiki.org/wiki/Open_Extension_Space_is_Compact | https://proofwiki.org/wiki/Open_Extension_Space_is_Compact | [
"Open Extension Topologies",
"Examples of Compact Topological Spaces"
] | [
"Definition:Topological Space",
"Definition:Open Extension Topology",
"Definition:Compact Topological Space"
] | [
"Definition:Open Extension Topology",
"Definition:Open Set/Topology",
"Definition:Open Cover",
"Definition:Subcover",
"Definition:Cover of Set/Finite",
"Definition:Compact Topological Space"
] |
proofwiki-3808 | Open Extension Space is Connected | Let $T = \struct {S, \tau}$ be a topological space.
Let $T^*_{\bar p} = \struct {S^*_p, \tau^*_{\bar p} }$ be the open extension space of $T$.
Then $T^*_{\bar p}$ is a connected space. | The only open set of $T$ which contains $p$ is $S^*_p$.
Therefore it is impossible to set up a separation of $T$, as $S^*_p$ will always need to be an element of such a separation.
{{qed}} | Let $T = \struct {S, \tau}$ be a [[Definition:Topological Space|topological space]].
Let $T^*_{\bar p} = \struct {S^*_p, \tau^*_{\bar p} }$ be the [[Definition:Open Extension Space|open extension space]] of $T$.
Then $T^*_{\bar p}$ is a [[Definition:Connected Topological Space|connected space]]. | The only [[Definition:Open Set (Topology)|open set]] of $T$ which contains $p$ is $S^*_p$.
Therefore it is impossible to set up a [[Definition:Separation (Topology)|separation]] of $T$, as $S^*_p$ will always need to be an [[Definition:Element|element]] of such a [[Definition:Separation (Topology)|separation]].
{{qed}... | Open Extension Space is Connected | https://proofwiki.org/wiki/Open_Extension_Space_is_Connected | https://proofwiki.org/wiki/Open_Extension_Space_is_Connected | [
"Open Extension Topologies",
"Examples of Connected Topological Spaces"
] | [
"Definition:Topological Space",
"Definition:Open Extension Topology",
"Definition:Connected Topological Space"
] | [
"Definition:Open Set/Topology",
"Definition:Separation (Topology)",
"Definition:Element",
"Definition:Separation (Topology)"
] |
proofwiki-3809 | Excluded Point Space is Connected | Let $T = \struct {S, \tau_{\bar p} }$ be an excluded point space.
Then $T^*_{\bar p}$ is a connected space. | We have:
: Excluded Point Topology is Open Extension Topology of Discrete Topology
: Open Extension Space is Connected
{{qed}} | Let $T = \struct {S, \tau_{\bar p} }$ be an [[Definition:Excluded Point Space|excluded point space]].
Then $T^*_{\bar p}$ is a [[Definition:Connected Topological Space|connected space]]. | We have:
: [[Excluded Point Topology is Open Extension Topology of Discrete Topology]]
: [[Open Extension Space is Connected]]
{{qed}} | Excluded Point Space is Connected/Proof 1 | https://proofwiki.org/wiki/Excluded_Point_Space_is_Connected | https://proofwiki.org/wiki/Excluded_Point_Space_is_Connected/Proof_1 | [
"Excluded Point Space is Connected",
"Excluded Point Topologies",
"Examples of Connected Topological Spaces"
] | [
"Definition:Excluded Point Topology",
"Definition:Connected Topological Space"
] | [
"Excluded Point Topology is Open Extension Topology of Discrete Topology",
"Open Extension Space is Connected"
] |
proofwiki-3810 | Excluded Point Space is Connected | Let $T = \struct {S, \tau_{\bar p} }$ be an excluded point space.
Then $T^*_{\bar p}$ is a connected space. | The only open set of $T$ which contains $p$ is $S$.
Therefore it is impossible to set up a separation of $T$, as $S$ will always need to be an element of such a separation.
{{qed}} | Let $T = \struct {S, \tau_{\bar p} }$ be an [[Definition:Excluded Point Space|excluded point space]].
Then $T^*_{\bar p}$ is a [[Definition:Connected Topological Space|connected space]]. | The only [[Definition:Open Set (Topology)|open set]] of $T$ which contains $p$ is $S$.
Therefore it is impossible to set up a [[Definition:Separation (Topology)|separation]] of $T$, as $S$ will always need to be an [[Definition:Element|element]] of such a [[Definition:Separation (Topology)|separation]].
{{qed}} | Excluded Point Space is Connected/Proof 2 | https://proofwiki.org/wiki/Excluded_Point_Space_is_Connected | https://proofwiki.org/wiki/Excluded_Point_Space_is_Connected/Proof_2 | [
"Excluded Point Space is Connected",
"Excluded Point Topologies",
"Examples of Connected Topological Spaces"
] | [
"Definition:Excluded Point Topology",
"Definition:Connected Topological Space"
] | [
"Definition:Open Set/Topology",
"Definition:Separation (Topology)",
"Definition:Element",
"Definition:Separation (Topology)"
] |
proofwiki-3811 | Excluded Point Space is Connected | Let $T = \struct {S, \tau_{\bar p} }$ be an excluded point space.
Then $T^*_{\bar p}$ is a connected space. | : Excluded Point Space is Ultraconnected
: Ultraconnected Space is Path-Connected
: Path-Connected Space is Connected
{{qed}} | Let $T = \struct {S, \tau_{\bar p} }$ be an [[Definition:Excluded Point Space|excluded point space]].
Then $T^*_{\bar p}$ is a [[Definition:Connected Topological Space|connected space]]. | : [[Excluded Point Space is Ultraconnected]]
: [[Ultraconnected Space is Path-Connected]]
: [[Path-Connected Space is Connected]]
{{qed}} | Excluded Point Space is Connected/Proof 3 | https://proofwiki.org/wiki/Excluded_Point_Space_is_Connected | https://proofwiki.org/wiki/Excluded_Point_Space_is_Connected/Proof_3 | [
"Excluded Point Space is Connected",
"Excluded Point Topologies",
"Examples of Connected Topological Spaces"
] | [
"Definition:Excluded Point Topology",
"Definition:Connected Topological Space"
] | [
"Excluded Point Space is Ultraconnected",
"Ultraconnected Space is Path-Connected",
"Path-Connected Space is Connected"
] |
proofwiki-3812 | Open Extension Space is Ultraconnected | Let $T = \struct {S, \tau}$ be a topological space.
Let $T^*_{\bar p} = \struct {S^*_p, \tau^*_{\bar p} }$ be the open extension space of $T$.
Then $T^*_{\bar p}$ is ultraconnected. | Apart from $S^*_p$, every open set of $T^*_{\bar p}$ does not contain $p$, by definition of open extension space.
So, apart from $\O$, every closed set of $T$ does contain $p$, by definition of closed set.
So every pair of closed sets of $T$ has an intersection which contains at least $p$.
So there are no non-empty dis... | Let $T = \struct {S, \tau}$ be a [[Definition:Topological Space|topological space]].
Let $T^*_{\bar p} = \struct {S^*_p, \tau^*_{\bar p} }$ be the [[Definition:Open Extension Space|open extension space]] of $T$.
Then $T^*_{\bar p}$ is [[Definition:Ultraconnected Space|ultraconnected]]. | Apart from $S^*_p$, every [[Definition:Open Set (Topology)|open set]] of $T^*_{\bar p}$ does not contain $p$, by definition of [[Definition:Open Extension Space|open extension space]].
So, apart from $\O$, every [[Definition:Closed Set (Topology)|closed set]] of $T$ does contain $p$, by definition of [[Definition:Clos... | Open Extension Space is Ultraconnected | https://proofwiki.org/wiki/Open_Extension_Space_is_Ultraconnected | https://proofwiki.org/wiki/Open_Extension_Space_is_Ultraconnected | [
"Open Extension Topologies",
"Examples of Ultraconnected Spaces"
] | [
"Definition:Topological Space",
"Definition:Open Extension Topology",
"Definition:Ultraconnected Space"
] | [
"Definition:Open Set/Topology",
"Definition:Open Extension Topology",
"Definition:Closed Set/Topology",
"Definition:Closed Set/Topology",
"Definition:Closed Set/Topology",
"Definition:Set Intersection",
"Definition:Non-Empty Set",
"Definition:Disjoint Sets",
"Definition:Closed Set/Topology",
"Defi... |
proofwiki-3813 | Excluded Point Space is Ultraconnected | Let $T = \struct {S, \tau_{\bar p} }$ be an excluded point space.
Then $T$ is ultraconnected. | : Excluded Point Topology is Open Extension Topology of Discrete Topology
: Open Extension Space is Ultraconnected
{{qed}} | Let $T = \struct {S, \tau_{\bar p} }$ be an [[Definition:Excluded Point Space|excluded point space]].
Then $T$ is [[Definition:Ultraconnected Space|ultraconnected]]. | : [[Excluded Point Topology is Open Extension Topology of Discrete Topology]]
: [[Open Extension Space is Ultraconnected]]
{{qed}} | Excluded Point Space is Ultraconnected/Proof 1 | https://proofwiki.org/wiki/Excluded_Point_Space_is_Ultraconnected | https://proofwiki.org/wiki/Excluded_Point_Space_is_Ultraconnected/Proof_1 | [
"Excluded Point Space is Ultraconnected",
"Excluded Point Topologies",
"Examples of Ultraconnected Spaces"
] | [
"Definition:Excluded Point Topology",
"Definition:Ultraconnected Space"
] | [
"Excluded Point Topology is Open Extension Topology of Discrete Topology",
"Open Extension Space is Ultraconnected"
] |
proofwiki-3814 | Excluded Point Space is Ultraconnected | Let $T = \struct {S, \tau_{\bar p} }$ be an excluded point space.
Then $T$ is ultraconnected. | Apart from $S$, every open set of $T$ does not contain $p$, by definition of excluded point space.
So, apart from $\O$, every closed set of $T$ does contain $p$, by definition of closed set.
So every pair of closed sets of $T$ has an intersection which contains at least $p$.
So there are no non-empty disjoint closed se... | Let $T = \struct {S, \tau_{\bar p} }$ be an [[Definition:Excluded Point Space|excluded point space]].
Then $T$ is [[Definition:Ultraconnected Space|ultraconnected]]. | Apart from $S$, every [[Definition:Open Set (Topology)|open set]] of $T$ does not contain $p$, by definition of [[Definition:Excluded Point Space|excluded point space]].
So, apart from $\O$, every [[Definition:Closed Set (Topology)|closed set]] of $T$ does contain $p$, by definition of [[Definition:Closed Set (Topolog... | Excluded Point Space is Ultraconnected/Proof 2 | https://proofwiki.org/wiki/Excluded_Point_Space_is_Ultraconnected | https://proofwiki.org/wiki/Excluded_Point_Space_is_Ultraconnected/Proof_2 | [
"Excluded Point Space is Ultraconnected",
"Excluded Point Topologies",
"Examples of Ultraconnected Spaces"
] | [
"Definition:Excluded Point Topology",
"Definition:Ultraconnected Space"
] | [
"Definition:Open Set/Topology",
"Definition:Excluded Point Topology",
"Definition:Closed Set/Topology",
"Definition:Closed Set/Topology",
"Definition:Closed Set/Topology",
"Definition:Set Intersection",
"Definition:Non-Empty Set",
"Definition:Disjoint Sets",
"Definition:Closed Set/Topology",
"Defi... |
proofwiki-3815 | Excluded Point Space is not Irreducible | Let $T = \struct {S, \tau_{\bar p} }$ be an excluded point space with at least three points.
Then $T^*_{\bar p}$ is not irreducible. | By definition, open sets of $S$ are precisely the open sets of $S \setminus \set p$ under the discrete topology.
Let $x, y \in S \setminus \set p: x \ne y$.
Then $\set x$ and $\set y$ are both open sets of $T$ such that $\set x \cap \set y = \O$.
Hence the result, by definition of irreducible.
{{qed}} | Let $T = \struct {S, \tau_{\bar p} }$ be an [[Definition:Excluded Point Topology|excluded point space]] with at least three [[Definition:Point of Set|points]].
Then $T^*_{\bar p}$ is not [[Definition:Irreducible Space|irreducible]]. | By definition, [[Definition:Open Set (Topology)|open sets]] of $S$ are precisely the [[Definition:Open Set (Topology)|open sets]] of $S \setminus \set p$ under the [[Definition:Discrete Topology|discrete topology]].
Let $x, y \in S \setminus \set p: x \ne y$.
Then $\set x$ and $\set y$ are both [[Definition:Open Set ... | Excluded Point Space is not Irreducible | https://proofwiki.org/wiki/Excluded_Point_Space_is_not_Irreducible | https://proofwiki.org/wiki/Excluded_Point_Space_is_not_Irreducible | [
"Excluded Point Topologies",
"Examples of Irreducible Spaces"
] | [
"Definition:Excluded Point Topology",
"Definition:Element",
"Definition:Irreducible Space"
] | [
"Definition:Open Set/Topology",
"Definition:Open Set/Topology",
"Definition:Discrete Topology",
"Definition:Open Set/Topology",
"Definition:Irreducible Space"
] |
proofwiki-3816 | Excluded Point Space is Path-Connected | Let $T = \struct {S, \tau_{\bar p} }$ be an excluded point space.
Then $T^*_{\bar p}$ is path-connected. | : Excluded Point Topology is Open Extension Topology of Discrete Topology
: Open Extension Space is Path-Connected
{{qed}} | Let $T = \struct {S, \tau_{\bar p} }$ be an [[Definition:Excluded Point Topology|excluded point space]].
Then $T^*_{\bar p}$ is [[Definition:Path-Connected Space|path-connected]]. | : [[Excluded Point Topology is Open Extension Topology of Discrete Topology]]
: [[Open Extension Space is Path-Connected]]
{{qed}} | Excluded Point Space is Path-Connected/Proof 1 | https://proofwiki.org/wiki/Excluded_Point_Space_is_Path-Connected | https://proofwiki.org/wiki/Excluded_Point_Space_is_Path-Connected/Proof_1 | [
"Excluded Point Space is Path-Connected",
"Excluded Point Topologies",
"Examples of Path-Connected Spaces"
] | [
"Definition:Excluded Point Topology",
"Definition:Path-Connected/Topological Space"
] | [
"Excluded Point Topology is Open Extension Topology of Discrete Topology",
"Open Extension Space is Path-Connected"
] |
proofwiki-3817 | Excluded Point Space is Path-Connected | Let $T = \struct {S, \tau_{\bar p} }$ be an excluded point space.
Then $T^*_{\bar p}$ is path-connected. | :Excluded Point Space is Ultraconnected
:Ultraconnected Space is Path-Connected
{{qed}} | Let $T = \struct {S, \tau_{\bar p} }$ be an [[Definition:Excluded Point Topology|excluded point space]].
Then $T^*_{\bar p}$ is [[Definition:Path-Connected Space|path-connected]]. | :[[Excluded Point Space is Ultraconnected]]
:[[Ultraconnected Space is Path-Connected]]
{{qed}} | Excluded Point Space is Path-Connected/Proof 2 | https://proofwiki.org/wiki/Excluded_Point_Space_is_Path-Connected | https://proofwiki.org/wiki/Excluded_Point_Space_is_Path-Connected/Proof_2 | [
"Excluded Point Space is Path-Connected",
"Excluded Point Topologies",
"Examples of Path-Connected Spaces"
] | [
"Definition:Excluded Point Topology",
"Definition:Path-Connected/Topological Space"
] | [
"Excluded Point Space is Ultraconnected",
"Ultraconnected Space is Path-Connected"
] |
proofwiki-3818 | Open Extension Space is Path-Connected | Let $T = \struct {S, \tau}$ be a topological space.
Let $T^*_{\bar p} = \struct {S^*_p, \tau^*_{\bar p} }$ be the open extension space of $T$.
Then $T^*_{\bar p}$ is path-connected. | :Open Extension Space is Ultraconnected
:Ultraconnected Space is Path-Connected
{{qed}}
Category:Open Extension Topologies
Category:Examples of Path-Connected Spaces
ls9y9zz33kxfc0k95nub1jideqsdpj9 | Let $T = \struct {S, \tau}$ be a [[Definition:Topological Space|topological space]].
Let $T^*_{\bar p} = \struct {S^*_p, \tau^*_{\bar p} }$ be the [[Definition:Open Extension Space|open extension space]] of $T$.
Then $T^*_{\bar p}$ is [[Definition:Path-Connected Space|path-connected]]. | :[[Open Extension Space is Ultraconnected]]
:[[Ultraconnected Space is Path-Connected]]
{{qed}}
[[Category:Open Extension Topologies]]
[[Category:Examples of Path-Connected Spaces]]
ls9y9zz33kxfc0k95nub1jideqsdpj9 | Open Extension Space is Path-Connected | https://proofwiki.org/wiki/Open_Extension_Space_is_Path-Connected | https://proofwiki.org/wiki/Open_Extension_Space_is_Path-Connected | [
"Open Extension Topologies",
"Examples of Path-Connected Spaces"
] | [
"Definition:Topological Space",
"Definition:Open Extension Topology",
"Definition:Path-Connected/Topological Space"
] | [
"Open Extension Space is Ultraconnected",
"Ultraconnected Space is Path-Connected",
"Category:Open Extension Topologies",
"Category:Examples of Path-Connected Spaces"
] |
proofwiki-3819 | Excluded Point Space is not Injectively Path-Connected | Let $S$ be a set with at least two distinct elements.
Let $T = \struct {S, \tau_{\bar p} }$ be an excluded point space.
Then $T$ is not injectively path-connected. | Let $q \in S$ be such that $q \ne p$.
Let $f: \closedint 0 1 \to T$ be an injection such that:
{{begin-eqn}}
{{eqn | l = \map f 0
| r = p
}}
{{eqn | l = \map f 1
| r = q
}}
{{end-eqn}}
Because $f$ is an injection, it must be that:
:$f^{-1} \sqbrk {\set q} = \set 1$
where $f^{-1}$ denotes the preimage under ... | Let $S$ be a [[Definition:Set|set]] with at least two [[Definition:Distinct Elements|distinct elements]].
Let $T = \struct {S, \tau_{\bar p} }$ be an [[Definition:Excluded Point Topology|excluded point space]].
Then $T$ is not [[Definition:Injectively Path-Connected Space|injectively path-connected]]. | Let $q \in S$ be such that $q \ne p$.
Let $f: \closedint 0 1 \to T$ be an [[Definition:Injection|injection]] such that:
{{begin-eqn}}
{{eqn | l = \map f 0
| r = p
}}
{{eqn | l = \map f 1
| r = q
}}
{{end-eqn}}
Because $f$ is an [[Definition:Injection|injection]], it must be that:
:$f^{-1} \sqbrk {\set q... | Excluded Point Space is not Injectively Path-Connected | https://proofwiki.org/wiki/Excluded_Point_Space_is_not_Injectively_Path-Connected | https://proofwiki.org/wiki/Excluded_Point_Space_is_not_Injectively_Path-Connected | [
"Excluded Point Topologies",
"Examples of Injectively Path-Connected Spaces"
] | [
"Definition:Set",
"Definition:Distinct/Plural",
"Definition:Excluded Point Topology",
"Definition:Injectively Path-Connected/Topological Space"
] | [
"Definition:Injection",
"Definition:Injection",
"Definition:Preimage/Mapping/Subset",
"Definition:Open Set/Topology",
"Definition:Excluded Point Topology",
"Closed Real Interval is not Open Set",
"Definition:Open Set/Topology",
"Definition:Open Set/Topology",
"Definition:Preimage/Mapping/Subset",
... |
proofwiki-3820 | Basis for Excluded Point Space | Let $T = \struct {S, \tau_{\bar p} }$ be an excluded point space.
Consider the set $\BB$ defined as:
:$\BB = \set {\set x: x \in S \setminus \set p} \cup \set S$
Then $B$ is a basis for $S$. | Let $H \in \tau_{\bar p}$ be open in $T$.
If $H = S$ then trivially $H$ is the union of elements of $\BB$.
So suppose $H \ne S$.
Then by definition $p \notin H$ and so:
:$\forall y \in H: \exists \set y \in \BB$
Thus:
:$\ds H = \bigcup_{y \mathop \in H} \set y$
So $\BB$ is an analytic basis for $T$.
{{qed}}
It could eq... | Let $T = \struct {S, \tau_{\bar p} }$ be an [[Definition:Excluded Point Topology|excluded point space]].
Consider the set $\BB$ defined as:
:$\BB = \set {\set x: x \in S \setminus \set p} \cup \set S$
Then $B$ is a [[Definition:Basis (Topology)|basis]] for $S$. | Let $H \in \tau_{\bar p}$ be [[Definition:Open Set (Topology)|open]] in $T$.
If $H = S$ then trivially $H$ is the [[Definition:Set Union|union]] of elements of $\BB$.
So suppose $H \ne S$.
Then by definition $p \notin H$ and so:
:$\forall y \in H: \exists \set y \in \BB$
Thus:
:$\ds H = \bigcup_{y \mathop \in H} \s... | Basis for Excluded Point Space | https://proofwiki.org/wiki/Basis_for_Excluded_Point_Space | https://proofwiki.org/wiki/Basis_for_Excluded_Point_Space | [
"Excluded Point Topologies",
"Topological Bases"
] | [
"Definition:Excluded Point Topology",
"Definition:Basis (Topology)"
] | [
"Definition:Open Set/Topology",
"Definition:Set Union",
"Definition:Basis (Topology)/Analytic Basis",
"Definition:Basis (Topology)/Synthetic Basis",
"Category:Excluded Point Topologies",
"Category:Topological Bases"
] |
proofwiki-3821 | Excluded Point Space is Locally Path-Connected | Let $T = \struct {S, \tau_{\bar p} }$ be an excluded point space.
Then $T$ is locally path-connected. | Consider the set $\BB$ defined as:
:$\BB = \set {\set x: x \in S \setminus \set p} \cup \set S$
Then $\BB$ is a basis for $T$.
Let $H \in \BB$.
Then $\exists x \in S: H = \set x$.
From Point is Path-Connected to Itself we have that $H$ is path-connected.
Now consider the open set $S \in \BB$.
Let $a, b \in S$.
Let $\ma... | Let $T = \struct {S, \tau_{\bar p} }$ be an [[Definition:Excluded Point Topology|excluded point space]].
Then $T$ is [[Definition:Locally Path-Connected Space|locally path-connected]]. | Consider the set $\BB$ defined as:
:$\BB = \set {\set x: x \in S \setminus \set p} \cup \set S$
Then $\BB$ is a [[Basis for Excluded Point Space|basis for $T$]].
Let $H \in \BB$.
Then $\exists x \in S: H = \set x$.
From [[Point is Path-Connected to Itself]] we have that $H$ is [[Definition:Path-Connected Set|path-... | Excluded Point Space is Locally Path-Connected | https://proofwiki.org/wiki/Excluded_Point_Space_is_Locally_Path-Connected | https://proofwiki.org/wiki/Excluded_Point_Space_is_Locally_Path-Connected | [
"Excluded Point Topologies",
"Examples of Locally Path-Connected Spaces"
] | [
"Definition:Excluded Point Topology",
"Definition:Locally Path-Connected Space"
] | [
"Basis for Excluded Point Space",
"Point is Path-Connected to Itself",
"Definition:Path-Connected/Set",
"Definition:Open Set/Topology",
"Definition:Real Interval/Unit Interval/Closed",
"Definition:Mapping",
"Definition:Open Set/Topology",
"Definition:Path-Connected/Set",
"Definition:Basis (Topology)... |
proofwiki-3822 | Excluded Point Space is not Locally Injectively Path-Connected | Let $T = \struct {S, \tau_{\bar p} }$ be an excluded point space.
Then $T$ is not locally injectively path-connected. | Let $\BB \subseteq \tau_{\bar p}$ be a basis for $\tau_{\bar p}$.
Since $\BB$ covers $S$, there must be an open set $B \in \BB$ such that $p \in B$.
By definition of the excluded point topology, the only open set containing $p$ is $S$ itself.
Hence necessarily $S \in \BB$.
But by Excluded Point Space is not Injectively... | Let $T = \struct {S, \tau_{\bar p} }$ be an [[Definition:Excluded Point Topology|excluded point space]].
Then $T$ is not [[Definition:Locally Injectively Path-Connected Space|locally injectively path-connected]]. | Let $\BB \subseteq \tau_{\bar p}$ be a [[Definition:Basis (Topology)|basis]] for $\tau_{\bar p}$.
Since $\BB$ [[Definition:Cover of Set|covers]] $S$, there must be an [[Definition:Open Set (Topology)|open set]] $B \in \BB$ such that $p \in B$.
By definition of the [[Definition:Excluded Point Topology|excluded point t... | Excluded Point Space is not Locally Injectively Path-Connected | https://proofwiki.org/wiki/Excluded_Point_Space_is_not_Locally_Injectively_Path-Connected | https://proofwiki.org/wiki/Excluded_Point_Space_is_not_Locally_Injectively_Path-Connected | [
"Excluded Point Topologies",
"Examples of Locally Injectively Path-Connected Spaces"
] | [
"Definition:Excluded Point Topology",
"Definition:Locally Injectively Path-Connected Space"
] | [
"Definition:Basis (Topology)",
"Definition:Cover of Set",
"Definition:Open Set/Topology",
"Definition:Excluded Point Topology",
"Definition:Open Set/Topology",
"Excluded Point Space is not Injectively Path-Connected",
"Definition:Injectively Path-Connected/Topological Space",
"Definition:Injectively P... |
proofwiki-3823 | Partition Topology is Zero Dimensional | Let $T = \struct {S, \tau}$ be a partition space.
Then $T$ is zero dimensional. | Let $\PP$ be the partition which is the basis for $T$.
From Subset of Partition Space is Open iff Closed, all the elements of $\PP$ are both closed and open.
Hence the result, by definition of zero dimensional space
{{qed}}
Category:Partition Topologies
Category:Examples of Zero Dimensional Spaces
254s1m6npgzba25eiz5gw... | Let $T = \struct {S, \tau}$ be a [[Definition:Partition Space|partition space]].
Then $T$ is [[Definition:Zero Dimensional Space|zero dimensional]]. | Let $\PP$ be the [[Definition:Partition (Set Theory)|partition]] which is the [[Basis for Partition Topology|basis for $T$]].
From [[Subset of Partition Space is Open iff Closed]], all the elements of $\PP$ are [[Definition:Clopen Set|both closed and open]].
Hence the result, by definition of [[Definition:Zero Dimens... | Partition Topology is Zero Dimensional | https://proofwiki.org/wiki/Partition_Topology_is_Zero_Dimensional | https://proofwiki.org/wiki/Partition_Topology_is_Zero_Dimensional | [
"Partition Topologies",
"Examples of Zero Dimensional Spaces"
] | [
"Definition:Partition Topology",
"Definition:Zero Dimensional Space"
] | [
"Definition:Set Partition",
"Basis for Partition Topology",
"Subset of Partition Space is Open iff Closed",
"Definition:Clopen Set",
"Definition:Zero Dimensional Space",
"Category:Partition Topologies",
"Category:Examples of Zero Dimensional Spaces"
] |
proofwiki-3824 | Discrete Space is Zero Dimensional | Let $T = \struct {S, \tau}$ be a discrete space.
Then $T$ is zero dimensional. | We have from Partition of Singletons yields Discrete Topology that a discrete space is a partition space.
The result follows from Partition Topology is Zero Dimensional.
{{qed}} | Let $T = \struct {S, \tau}$ be a [[Definition:Discrete Space|discrete space]].
Then $T$ is [[Definition:Zero Dimensional Space|zero dimensional]]. | We have from [[Partition of Singletons yields Discrete Topology]] that a [[Definition:Discrete Space|discrete space]] is a [[Definition:Partition Space|partition space]].
The result follows from [[Partition Topology is Zero Dimensional]].
{{qed}} | Discrete Space is Zero Dimensional/Proof 1 | https://proofwiki.org/wiki/Discrete_Space_is_Zero_Dimensional | https://proofwiki.org/wiki/Discrete_Space_is_Zero_Dimensional/Proof_1 | [
"Discrete Space is Zero Dimensional",
"Discrete Topologies",
"Examples of Zero Dimensional Spaces",
"Sequence of Implications of Disconnectedness Properties"
] | [
"Definition:Discrete Topology",
"Definition:Zero Dimensional Space"
] | [
"Partition of Singletons yields Discrete Topology",
"Definition:Discrete Topology",
"Definition:Partition Topology",
"Partition Topology is Zero Dimensional"
] |
proofwiki-3825 | Discrete Space is Zero Dimensional | Let $T = \struct {S, \tau}$ be a discrete space.
Then $T$ is zero dimensional. | Let $\BB$ be the set:
:$\BB := \set {\set x: x \in S}$
From Basis for Discrete Topology, $\BB$ is a basis for $T$.
From Set in Discrete Topology is Clopen, all the elements of $\BB$ are both closed and open.
Hence the result, by definition of zero dimensional space
{{qed}} | Let $T = \struct {S, \tau}$ be a [[Definition:Discrete Space|discrete space]].
Then $T$ is [[Definition:Zero Dimensional Space|zero dimensional]]. | Let $\BB$ be the set:
:$\BB := \set {\set x: x \in S}$
From [[Basis for Discrete Topology]], $\BB$ is a [[Definition:Basis (Topology)|basis]] for $T$.
From [[Set in Discrete Topology is Clopen]], all the elements of $\BB$ are [[Definition:Clopen Set|both closed and open]].
Hence the result, by definition of [[Defini... | Discrete Space is Zero Dimensional/Proof 2 | https://proofwiki.org/wiki/Discrete_Space_is_Zero_Dimensional | https://proofwiki.org/wiki/Discrete_Space_is_Zero_Dimensional/Proof_2 | [
"Discrete Space is Zero Dimensional",
"Discrete Topologies",
"Examples of Zero Dimensional Spaces",
"Sequence of Implications of Disconnectedness Properties"
] | [
"Definition:Discrete Topology",
"Definition:Zero Dimensional Space"
] | [
"Basis for Discrete Topology",
"Definition:Basis (Topology)",
"Set in Discrete Topology is Clopen",
"Definition:Clopen Set",
"Definition:Zero Dimensional Space"
] |
proofwiki-3826 | Discrete Space is Scattered | Let $T = \struct {S, \tau}$ be a topological space where $\tau$ is the discrete topology on $S$.
Then $T$ is a scattered space. | We have that Topological Space is Discrete iff All Points are Isolated.
So, by definition, no subset $H \subseteq S$ of $T$ such that $H \ne \O$ is dense-in-itself.
So, again, by definition, $T$ is scattered.
{{qed}} | Let $T = \struct {S, \tau}$ be a [[Definition:Topological Space|topological space]] where $\tau$ is the [[Definition:Discrete Topology|discrete topology]] on $S$.
Then $T$ is a [[Definition:Scattered Space|scattered space]]. | We have that [[Topological Space is Discrete iff All Points are Isolated]].
So, by definition, no [[Definition:Subset|subset]] $H \subseteq S$ of $T$ such that $H \ne \O$ is [[Definition:Dense-in-itself|dense-in-itself]].
So, again, by definition, $T$ is [[Definition:Scattered Space|scattered]].
{{qed}} | Discrete Space is Scattered | https://proofwiki.org/wiki/Discrete_Space_is_Scattered | https://proofwiki.org/wiki/Discrete_Space_is_Scattered | [
"Discrete Topologies",
"Examples of Scattered Spaces",
"Sequence of Implications of Disconnectedness Properties"
] | [
"Definition:Topological Space",
"Definition:Discrete Topology",
"Definition:Scattered Space"
] | [
"Topological Space is Discrete iff All Points are Isolated",
"Definition:Subset",
"Definition:Dense-in-itself",
"Definition:Scattered Space"
] |
proofwiki-3827 | Discrete Space is Extremally Disconnected Hausdorff | Let $T = \struct {S, \tau}$ be a topological space where $\tau$ is the discrete topology on $S$.
Then $T$ is an extremally disconnected Hausdorff space. | First we note that as Discrete Space satisfies all Separation Properties, $T$ is a $T_2$ (Hausdorff) space.
Then from Interior Equals Closure of Subset of Discrete Space, it follows directly that the closure of every open set of $T$ is open.
Hence, by definition, $T$ is extremally disconnected Hausdorff.
{{qed}} | Let $T = \struct {S, \tau}$ be a [[Definition:Topological Space|topological space]] where $\tau$ is the [[Definition:Discrete Topology|discrete topology]] on $S$.
Then $T$ is an [[Definition:Extremally Disconnected Hausdorff Space|extremally disconnected Hausdorff space]]. | First we note that as [[Discrete Space satisfies all Separation Properties]], $T$ is a [[Definition:Hausdorff Space|$T_2$ (Hausdorff) space]].
Then from [[Interior Equals Closure of Subset of Discrete Space]], it follows directly that the [[Definition:Closure (Topology)|closure]] of every [[Definition:Open Set (Topolo... | Discrete Space is Extremally Disconnected Hausdorff | https://proofwiki.org/wiki/Discrete_Space_is_Extremally_Disconnected_Hausdorff | https://proofwiki.org/wiki/Discrete_Space_is_Extremally_Disconnected_Hausdorff | [
"Discrete Topologies",
"Examples of Extremally Disconnected Hausdorff Spaces",
"Sequence of Implications of Disconnectedness Properties"
] | [
"Definition:Topological Space",
"Definition:Discrete Topology",
"Definition:Extremally Disconnected Hausdorff Space"
] | [
"Discrete Space satisfies all Separation Properties",
"Definition:T2 Space",
"Interior Equals Closure of Subset of Discrete Space",
"Definition:Closure (Topology)",
"Definition:Open Set/Topology",
"Definition:Open Set/Topology",
"Definition:Extremally Disconnected Hausdorff Space"
] |
proofwiki-3828 | Totally Separated Space is Totally Disconnected | Let $T = \struct {S, \tau}$ be a topological space which is totally separated.
Then $T$ is totally disconnected. | {{Recall|Totally Disconnected Space|totally disconnected space}}
{{:Definition:Totally Disconnected Space}}
Let $T = \struct {S, \tau}$ be a totally separated space.
{{Recall|Totally Separated Space|totally separated space}}
{{:Definition:Totally Separated Space/Definition 1}}
{{AimForCont}} $T$ were not totally discon... | Let $T = \struct {S, \tau}$ be a [[Definition:Topological Space|topological space]] which is [[Definition:Totally Separated Space|totally separated]].
Then $T$ is [[Definition:Totally Disconnected Space|totally disconnected]]. | {{Recall|Totally Disconnected Space|totally disconnected space}}
{{:Definition:Totally Disconnected Space}}
Let $T = \struct {S, \tau}$ be a [[Definition:Totally Separated Space|totally separated space]].
{{Recall|Totally Separated Space|totally separated space}}
{{:Definition:Totally Separated Space/Definition 1}}
... | Totally Separated Space is Totally Disconnected/Proof 1 | https://proofwiki.org/wiki/Totally_Separated_Space_is_Totally_Disconnected | https://proofwiki.org/wiki/Totally_Separated_Space_is_Totally_Disconnected/Proof_1 | [
"Totally Separated Space is Totally Disconnected",
"Totally Separated Spaces",
"Totally Disconnected Spaces",
"Sequence of Implications of Disconnectedness Properties"
] | [
"Definition:Topological Space",
"Definition:Totally Separated Space",
"Definition:Totally Disconnected Space"
] | [
"Definition:Totally Separated Space",
"Definition:Totally Disconnected Space",
"Definition:Connected Set (Topology)",
"Definition:Connected Set (Topology)",
"Definition:Separation (Topology)",
"Definition:Contradiction",
"Definition:Totally Separated Space",
"Proof by Contradiction"
] |
proofwiki-3829 | Open Extension Topology is not Perfectly T4 | Let $T = \struct {S, \tau}$ be a topological space.
Let $T^*_{\bar p} = \struct {S^*_p, \tau^*_{\bar p} }$ be the open extension space of $T$.
Then $T^*_{\bar p}$ is not a perfectly $T_4$ space. | By definition:
:$\tau^*_{\bar p} = \set {U: U \in \tau} \cup \set {S^*_p}$
We have that $S$ is an open set in $T$ and so open set in $T^*_{\bar p}$.
So $\set p = S^*_p \setminus S$ is closed in $T^*_{\bar p}$.
The only open set in $T^*_{\bar p}$ which contains $p$ is $S^*_p$.
So $\set p$ can not be the intersection of ... | Let $T = \struct {S, \tau}$ be a [[Definition:Topological Space|topological space]].
Let $T^*_{\bar p} = \struct {S^*_p, \tau^*_{\bar p} }$ be the [[Definition:Open Extension Space|open extension space]] of $T$.
Then $T^*_{\bar p}$ is not a [[Definition:Perfectly T4 Space|perfectly $T_4$ space]]. | By definition:
:$\tau^*_{\bar p} = \set {U: U \in \tau} \cup \set {S^*_p}$
We have that $S$ is an [[Definition:Open Set (Topology)|open set]] in $T$ and so [[Definition:Open Set (Topology)|open set]] in $T^*_{\bar p}$.
So $\set p = S^*_p \setminus S$ is [[Definition:Closed Set (Topology)|closed]] in $T^*_{\bar p}$.... | Open Extension Topology is not Perfectly T4 | https://proofwiki.org/wiki/Open_Extension_Topology_is_not_Perfectly_T4 | https://proofwiki.org/wiki/Open_Extension_Topology_is_not_Perfectly_T4 | [
"Open Extension Topologies",
"Examples of Perfectly T4 Spaces"
] | [
"Definition:Topological Space",
"Definition:Open Extension Topology",
"Definition:Perfectly T4 Space"
] | [
"Definition:Open Set/Topology",
"Definition:Open Set/Topology",
"Definition:Closed Set/Topology",
"Definition:Open Set/Topology",
"Definition:Set Intersection",
"Definition:Open Set/Topology",
"Definition:Set Intersection",
"Definition:Set Intersection/Countable Intersection",
"Definition:Perfectly ... |
proofwiki-3830 | Excluded Point Space is not Perfectly T4 | Let $T = \struct {S, \tau_{\bar p} }$ be an excluded point space.
Then $T$ is not a perfectly $T_4$ space. | :Excluded Point Topology is Open Extension Topology of Discrete Topology
:Open Extension Topology is not Perfectly T4
{{qed}} | Let $T = \struct {S, \tau_{\bar p} }$ be an [[Definition:Excluded Point Topology|excluded point space]].
Then $T$ is not a [[Definition:Perfectly T4 Space|perfectly $T_4$ space]]. | :[[Excluded Point Topology is Open Extension Topology of Discrete Topology]]
:[[Open Extension Topology is not Perfectly T4]]
{{qed}} | Excluded Point Space is not Perfectly T4 | https://proofwiki.org/wiki/Excluded_Point_Space_is_not_Perfectly_T4 | https://proofwiki.org/wiki/Excluded_Point_Space_is_not_Perfectly_T4 | [
"Excluded Point Topologies",
"Examples of Perfectly T4 Spaces"
] | [
"Definition:Excluded Point Topology",
"Definition:Perfectly T4 Space"
] | [
"Excluded Point Topology is Open Extension Topology of Discrete Topology",
"Open Extension Topology is not Perfectly T4"
] |
proofwiki-3831 | Type is Realized in some Elementary Extension | Let $\MM$ be an $\LL$-structure.
Let $A$ be a subset of the universe of $\MM$.
Let $p$ be an $n$-type over $A$.
There exists an elementary extension of $\MM$ which realizes $p$. | The idea is to work in a language with constant symbols for all elements of $\MM$ and show that the union of $p$ and the elementary diagram of $\MM$ is satisfiable.
Since $\MM$ naturally embeds into any model of such a theory, this will prove the theorem.
Let $\LL_\MM$ be the language obtained by adding to $\LL$ consta... | Let $\MM$ be an [[Definition:First-Order Structure|$\LL$-structure]].
Let $A$ be a subset of the universe of $\MM$.
Let $p$ be an [[Definition:Type|$n$-type]] over $A$.
There exists an [[Definition:Elementary Extension|elementary extension]] of $\MM$ which [[Definition:Realization of Type|realizes]] $p$. | The idea is to work in a language with constant symbols for all elements of $\MM$ and show that the union of $p$ and the [[Definition:Elementary Diagram|elementary diagram]] of $\MM$ is satisfiable.
Since $\MM$ naturally embeds into any model of such a theory, this will prove the theorem.
Let $\LL_\MM$ be the langua... | Type is Realized in some Elementary Extension | https://proofwiki.org/wiki/Type_is_Realized_in_some_Elementary_Extension | https://proofwiki.org/wiki/Type_is_Realized_in_some_Elementary_Extension | [
"Model Theory for Predicate Logic"
] | [
"Definition:Structure for Predicate Logic",
"Definition:Type",
"Definition:Elementary Extension",
"Definition:Type"
] | [
"Definition:Elementary Diagram",
"Compactness Theorem",
"Compactness Theorem",
"Definition:Elementary Map (Model Theory)",
"Category:Model Theory for Predicate Logic"
] |
proofwiki-3832 | Subset of Excluded Point Space is not Dense-in-itself | Let $T = \struct {S, \tau_{\bar p} }$ be a excluded point space such that $S$ is not a singleton.
Let $H \subseteq S$.
Then $H$ is not dense-in-itself. | From Limit Points in Excluded Point Space, the only limit point of $H$ is $p$.
So by definition, all points of $H$ are isolated in $H$ except $p$.
So if $H \ne \set p$, $H$ contains at least one point which is isolated in $H$.
As for $p$ itself, from Singleton Point is Isolated we have that $p$ is itself isolated in $... | Let $T = \struct {S, \tau_{\bar p} }$ be a [[Definition:Excluded Point Topology|excluded point space]] such that $S$ is not a [[Definition:Singleton|singleton]].
Let $H \subseteq S$.
Then $H$ is not [[Definition:Dense-in-itself|dense-in-itself]]. | From [[Limit Points in Excluded Point Space]], the only [[Definition:Limit Point of Set|limit point]] of $H$ is $p$.
So by definition, all points of $H$ are [[Definition:Isolated Point of Subset|isolated in $H$]] except $p$.
So if $H \ne \set p$, $H$ contains at least one point which is [[Definition:Isolated Point o... | Subset of Excluded Point Space is not Dense-in-itself | https://proofwiki.org/wiki/Subset_of_Excluded_Point_Space_is_not_Dense-in-itself | https://proofwiki.org/wiki/Subset_of_Excluded_Point_Space_is_not_Dense-in-itself | [
"Excluded Point Topologies",
"Examples of Dense-in-itself"
] | [
"Definition:Excluded Point Topology",
"Definition:Singleton",
"Definition:Dense-in-itself"
] | [
"Limit Points in Excluded Point Space",
"Definition:Limit Point/Topology/Set",
"Definition:Isolated Point (Topology)/Subset",
"Definition:Isolated Point (Topology)/Subset",
"Singleton Point is Isolated",
"Definition:Isolated Point (Topology)/Subset",
"Definition:Isolated Point (Topology)/Subset",
"Def... |
proofwiki-3833 | Excluded Point Space is Scattered | Let $T = \struct {S, \tau_{\bar p} }$ be an excluded point space.
Then $T$ is a scattered space. | We have that Subset of Excluded Point Space is not Dense-in-itself.
So, by definition, $T$ is scattered.
{{qed}} | Let $T = \struct {S, \tau_{\bar p} }$ be an [[Definition:Excluded Point Topology|excluded point space]].
Then $T$ is a [[Definition:Scattered Space|scattered space]]. | We have that [[Subset of Excluded Point Space is not Dense-in-itself]].
So, by definition, $T$ is [[Definition:Scattered Space|scattered]].
{{qed}} | Excluded Point Space is Scattered | https://proofwiki.org/wiki/Excluded_Point_Space_is_Scattered | https://proofwiki.org/wiki/Excluded_Point_Space_is_Scattered | [
"Excluded Point Topologies",
"Examples of Scattered Spaces"
] | [
"Definition:Excluded Point Topology",
"Definition:Scattered Space"
] | [
"Subset of Excluded Point Space is not Dense-in-itself",
"Definition:Scattered Space"
] |
proofwiki-3834 | Dispersion Point of Excluded Point Space | Let $T = \struct {S, \tau_{\bar p} }$ be an excluded point space.
Then $p$ is a dispersion point of $T$. | We have that the Excluded Point Topology is Open Extension Topology of Discrete Topology.
So $S \setminus \set p$ is a discrete space.
Then a discrete space is totally disconnected.
The result follows from definition of dispersion point.
{{qed}} | Let $T = \struct {S, \tau_{\bar p} }$ be an [[Definition:Excluded Point Topology|excluded point space]].
Then $p$ is a [[Definition:Dispersion Point|dispersion point]] of $T$. | We have that the [[Excluded Point Topology is Open Extension Topology of Discrete Topology]].
So $S \setminus \set p$ is a [[Definition:Discrete Space|discrete space]].
Then a [[Totally Disconnected and Locally Connected Space is Discrete|discrete space is totally disconnected]].
The result follows from definition o... | Dispersion Point of Excluded Point Space | https://proofwiki.org/wiki/Dispersion_Point_of_Excluded_Point_Space | https://proofwiki.org/wiki/Dispersion_Point_of_Excluded_Point_Space | [
"Excluded Point Topologies",
"Examples of Dispersion Points"
] | [
"Definition:Excluded Point Topology",
"Definition:Dispersion Point"
] | [
"Excluded Point Topology is Open Extension Topology of Discrete Topology",
"Definition:Discrete Topology",
"Totally Disconnected and Locally Connected Space is Discrete",
"Definition:Dispersion Point"
] |
proofwiki-3835 | Excluded Point Space is First-Countable | Let $T = \struct {S, \tau_{\bar p} }$ be an excluded point space.
Then $T$ is first-countable. | Let $x \in S$ such that $x \ne p$.
By definition, the set $\set x$ is open in $T$.
Let $U \in \tau_{\bar p}: x \in U$.
Then $\set x \subseteq U$ and so $\set {\set x}$ is a local basis at $x$ which is trivially countable.
Now if $x = p$ there is only one $U \in \tau_{\bar p}: p \in U$, and that is $S$.
So $\set S$ is a... | Let $T = \struct {S, \tau_{\bar p} }$ be an [[Definition:Excluded Point Topology|excluded point space]].
Then $T$ is [[Definition:First-Countable Space|first-countable]]. | Let $x \in S$ such that $x \ne p$.
By definition, the set $\set x$ is [[Definition:Open Set (Topology)|open]] in $T$.
Let $U \in \tau_{\bar p}: x \in U$.
Then $\set x \subseteq U$ and so $\set {\set x}$ is a [[Definition:Local Basis|local basis]] at $x$ which is trivially [[Definition:Countable Set|countable]].
No... | Excluded Point Space is First-Countable | https://proofwiki.org/wiki/Excluded_Point_Space_is_First-Countable | https://proofwiki.org/wiki/Excluded_Point_Space_is_First-Countable | [
"Excluded Point Topologies",
"Examples of First-Countable Spaces"
] | [
"Definition:Excluded Point Topology",
"Definition:First-Countable Space"
] | [
"Definition:Open Set/Topology",
"Definition:Local Basis",
"Definition:Countable Set",
"Definition:Local Basis",
"Definition:Countable Set",
"Definition:First-Countable Space"
] |
proofwiki-3836 | Excluded Point Space is Sequentially Compact | Let $T = \struct {S, \tau_{\bar p} }$ be an excluded point space.
Then $T$ is a sequentially compact space. | :Excluded Point Space is Compact
:Compact Space is Countably Compact
:Excluded Point Space is First-Countable
:First-Countable Space is Sequentially Compact iff Countably Compact
{{qed}} | Let $T = \struct {S, \tau_{\bar p} }$ be an [[Definition:Excluded Point Topology|excluded point space]].
Then $T$ is a [[Definition:Sequentially Compact Space|sequentially compact space]]. | :[[Excluded Point Space is Compact]]
:[[Compact Space is Countably Compact]]
:[[Excluded Point Space is First-Countable]]
:[[First-Countable Space is Sequentially Compact iff Countably Compact]]
{{qed}} | Excluded Point Space is Sequentially Compact | https://proofwiki.org/wiki/Excluded_Point_Space_is_Sequentially_Compact | https://proofwiki.org/wiki/Excluded_Point_Space_is_Sequentially_Compact | [
"Excluded Point Topologies",
"Examples of Sequentially Compact Spaces"
] | [
"Definition:Excluded Point Topology",
"Definition:Sequentially Compact Space"
] | [
"Excluded Point Space is Compact",
"Compact Space is Countably Compact",
"Excluded Point Space is First-Countable",
"First-Countable Space is Sequentially Compact iff Countably Compact"
] |
proofwiki-3837 | Countable Excluded Point Space is Second-Countable | Let $T = \struct {S, \tau_{\bar p} }$ be a countable excluded point space.
Then $T$ is a second-countable space. | Consider the set $\BB$ defined as:
:$\BB = \set {\set x: x \in S \setminus \set p} \cup \set S$
From Basis for Excluded Point Space, $\BB$ is a basis for $T$, and trivially has the same cardinality as $S$.
So by definition, if $S$ is countable, then $T$ is second-countable.
{{qed}} | Let $T = \struct {S, \tau_{\bar p} }$ be a [[Definition:Countable Excluded Point Topology|countable excluded point space]].
Then $T$ is a [[Definition:Second-Countable Space|second-countable space]]. | Consider the set $\BB$ defined as:
:$\BB = \set {\set x: x \in S \setminus \set p} \cup \set S$
From [[Basis for Excluded Point Space]], $\BB$ is a [[Definition:Basis (Topology)|basis]] for $T$, and trivially has the same [[Definition:Cardinality|cardinality]] as $S$.
So by definition, if $S$ is [[Definition:Countabl... | Countable Excluded Point Space is Second-Countable/Proof 1 | https://proofwiki.org/wiki/Countable_Excluded_Point_Space_is_Second-Countable | https://proofwiki.org/wiki/Countable_Excluded_Point_Space_is_Second-Countable/Proof_1 | [
"Countable Excluded Point Space is Second-Countable",
"Countable Excluded Point Topologies",
"Examples of Second-Countable Spaces"
] | [
"Definition:Excluded Point Topology/Countable",
"Definition:Second-Countable Space"
] | [
"Basis for Excluded Point Space",
"Definition:Basis (Topology)",
"Definition:Cardinality",
"Definition:Countable Set",
"Definition:Second-Countable Space"
] |
proofwiki-3838 | Countable Excluded Point Space is Second-Countable | Let $T = \struct {S, \tau_{\bar p} }$ be a countable excluded point space.
Then $T$ is a second-countable space. | We have:
: Countable Discrete Space is Second-Countable
: Excluded Point Topology is Open Extension Topology of Discrete Topology
The result follows from Condition for Open Extension Space to be Second-Countable.
{{qed}} | Let $T = \struct {S, \tau_{\bar p} }$ be a [[Definition:Countable Excluded Point Topology|countable excluded point space]].
Then $T$ is a [[Definition:Second-Countable Space|second-countable space]]. | We have:
: [[Countable Discrete Space is Second-Countable]]
: [[Excluded Point Topology is Open Extension Topology of Discrete Topology]]
The result follows from [[Condition for Open Extension Space to be Second-Countable]].
{{qed}} | Countable Excluded Point Space is Second-Countable/Proof 2 | https://proofwiki.org/wiki/Countable_Excluded_Point_Space_is_Second-Countable | https://proofwiki.org/wiki/Countable_Excluded_Point_Space_is_Second-Countable/Proof_2 | [
"Countable Excluded Point Space is Second-Countable",
"Countable Excluded Point Topologies",
"Examples of Second-Countable Spaces"
] | [
"Definition:Excluded Point Topology/Countable",
"Definition:Second-Countable Space"
] | [
"Countable Discrete Space is Second-Countable",
"Excluded Point Topology is Open Extension Topology of Discrete Topology",
"Condition for Open Extension Space to be Second-Countable"
] |
proofwiki-3839 | Uncountable Excluded Point Space is not Second-Countable | Let $T = \struct {S, \tau_{\bar p} }$ be an uncountable excluded point space.
Then $T$ is not second-countable. | Let $H = S \setminus \left\{{p}\right\}$ where $\setminus$ denotes set difference.
By definition, $H$ is an uncountable discrete space.
The result follows from Uncountable Discrete Space is not Second-Countable.
{{qed}} | Let $T = \struct {S, \tau_{\bar p} }$ be an [[Definition:Uncountable Excluded Point Topology|uncountable excluded point space]].
Then $T$ is not [[Definition:Second-Countable Space|second-countable]]. | Let $H = S \setminus \left\{{p}\right\}$ where $\setminus$ denotes [[Definition:Set Difference|set difference]].
By definition, $H$ is an [[Definition:Uncountable Discrete Topology|uncountable discrete space]].
The result follows from [[Uncountable Discrete Space is not Second-Countable]].
{{qed}} | Uncountable Excluded Point Space is not Second-Countable/Proof 1 | https://proofwiki.org/wiki/Uncountable_Excluded_Point_Space_is_not_Second-Countable | https://proofwiki.org/wiki/Uncountable_Excluded_Point_Space_is_not_Second-Countable/Proof_1 | [
"Uncountable Excluded Point Space is not Second-Countable",
"Uncountable Excluded Point Topologies",
"Examples of Second-Countable Spaces"
] | [
"Definition:Excluded Point Topology/Uncountable",
"Definition:Second-Countable Space"
] | [
"Definition:Set Difference",
"Definition:Discrete Topology/Uncountable",
"Uncountable Discrete Space is not Second-Countable"
] |
proofwiki-3840 | Uncountable Excluded Point Space is not Second-Countable | Let $T = \struct {S, \tau_{\bar p} }$ be an uncountable excluded point space.
Then $T$ is not second-countable. | We have:
: Uncountable Discrete Space is not Second-Countable
: Excluded Point Topology is Open Extension Topology of Discrete Topology
The result follows from Condition for Open Extension Space to be Second-Countable
{{qed}} | Let $T = \struct {S, \tau_{\bar p} }$ be an [[Definition:Uncountable Excluded Point Topology|uncountable excluded point space]].
Then $T$ is not [[Definition:Second-Countable Space|second-countable]]. | We have:
: [[Uncountable Discrete Space is not Second-Countable]]
: [[Excluded Point Topology is Open Extension Topology of Discrete Topology]]
The result follows from [[Condition for Open Extension Space to be Second-Countable]]
{{qed}} | Uncountable Excluded Point Space is not Second-Countable/Proof 2 | https://proofwiki.org/wiki/Uncountable_Excluded_Point_Space_is_not_Second-Countable | https://proofwiki.org/wiki/Uncountable_Excluded_Point_Space_is_not_Second-Countable/Proof_2 | [
"Uncountable Excluded Point Space is not Second-Countable",
"Uncountable Excluded Point Topologies",
"Examples of Second-Countable Spaces"
] | [
"Definition:Excluded Point Topology/Uncountable",
"Definition:Second-Countable Space"
] | [
"Uncountable Discrete Space is not Second-Countable",
"Excluded Point Topology is Open Extension Topology of Discrete Topology",
"Condition for Open Extension Space to be Second-Countable"
] |
proofwiki-3841 | Countable Excluded Point Space is Separable | Let $T = \struct {S, \tau_{\bar p} }$ be a countable excluded point space.
Then $T$ is a separable space. | The closure of the set $S$ (trivially) equals $S$.
That is, $S$ is everywhere dense in $T$.
But as $S$ is countable it follows by definition that $T$ is separable.
{{qed}} | Let $T = \struct {S, \tau_{\bar p} }$ be a [[Definition:Countable Excluded Point Topology|countable excluded point space]].
Then $T$ is a [[Definition:Separable Space|separable space]]. | The [[Definition:Closure (Topology)|closure]] of the set $S$ (trivially) equals $S$.
That is, $S$ is [[Definition:Everywhere Dense|everywhere dense]] in $T$.
But as $S$ is [[Definition:Countable Set|countable]] it follows by definition that $T$ is [[Definition:Separable Space|separable]].
{{qed}} | Countable Excluded Point Space is Separable | https://proofwiki.org/wiki/Countable_Excluded_Point_Space_is_Separable | https://proofwiki.org/wiki/Countable_Excluded_Point_Space_is_Separable | [
"Countable Excluded Point Topologies",
"Examples of Separable Spaces"
] | [
"Definition:Excluded Point Topology/Countable",
"Definition:Separable Space"
] | [
"Definition:Closure (Topology)",
"Definition:Everywhere Dense",
"Definition:Countable Set",
"Definition:Separable Space"
] |
proofwiki-3842 | Uncountable Excluded Point Space is not Separable | Let $T = \struct {S, \tau_{\bar p} }$ be an uncountable excluded point space.
Then $T$ is not separable. | Let $H \subseteq S$ such that $H$ is countable.
Then $H \ne S$ as $S$ is uncountable by hypothesis.
From Limit Points in Excluded Point Space, the only limit point of $H$ is $p$.
So, by definition, the closure of $H$ is $H \cup \set p$.
From Countable Union of Countable Sets is Countable we have that $H \cup \set p$ is... | Let $T = \struct {S, \tau_{\bar p} }$ be an [[Definition:Uncountable Excluded Point Topology|uncountable excluded point space]].
Then $T$ is not [[Definition:Separable Space|separable]]. | Let $H \subseteq S$ such that $H$ is [[Definition:Countable Set|countable]].
Then $H \ne S$ as $S$ is [[Definition:Uncountable Set|uncountable]] by hypothesis.
From [[Limit Points in Excluded Point Space]], the only [[Definition:Limit Point of Set|limit point]] of $H$ is $p$.
So, by definition, the [[Definition:Clos... | Uncountable Excluded Point Space is not Separable | https://proofwiki.org/wiki/Uncountable_Excluded_Point_Space_is_not_Separable | https://proofwiki.org/wiki/Uncountable_Excluded_Point_Space_is_not_Separable | [
"Uncountable Excluded Point Topologies",
"Examples of Separable Spaces"
] | [
"Definition:Excluded Point Topology/Uncountable",
"Definition:Separable Space"
] | [
"Definition:Countable Set",
"Definition:Uncountable/Set",
"Limit Points in Excluded Point Space",
"Definition:Limit Point/Topology/Set",
"Definition:Closure (Topology)",
"Countable Union of Countable Sets is Countable",
"Definition:Countable Set",
"Definition:Everywhere Dense",
"Definition:Countable... |
proofwiki-3843 | Excluded Set Topology is Topology | Let $T = \struct {S, \tau_{\bar H} }$ be an excluded set space.
Then $\tau_{\bar H}$ is a topology on $S$, and $T$ is a topological space. | We have by definition that $S \in \tau_{\bar H}$.
Also, as $H \cap \O = \O$, we have that $\O \in \tau_{\bar H}$.
Now let $U_1, U_2 \in \tau_{\bar H}$.
By definition:
:$H \cap U_1 = \O$
and:
:$H \cap U_2 = \O$
and so by definition of set intersection:
:$H \cap \paren {U_1 \cap U_2} = \O$
So:
:$U_1 \cap U_2 \in \tau_{\b... | Let $T = \struct {S, \tau_{\bar H} }$ be an [[Definition:Excluded Set Space|excluded set space]].
Then $\tau_{\bar H}$ is a [[Definition:Topology|topology]] on $S$, and $T$ is a [[Definition:Topological Space|topological space]]. | We have [[Definition:Excluded Set Space|by definition]] that $S \in \tau_{\bar H}$.
Also, as $H \cap \O = \O$, we have that $\O \in \tau_{\bar H}$.
Now let $U_1, U_2 \in \tau_{\bar H}$.
By definition:
:$H \cap U_1 = \O$
and:
:$H \cap U_2 = \O$
and so by definition of [[Definition:Set Intersection|set intersection]]... | Excluded Set Topology is Topology | https://proofwiki.org/wiki/Excluded_Set_Topology_is_Topology | https://proofwiki.org/wiki/Excluded_Set_Topology_is_Topology | [
"Excluded Set Topologies"
] | [
"Definition:Excluded Set Topology",
"Definition:Topology",
"Definition:Topological Space"
] | [
"Definition:Excluded Set Topology",
"Definition:Set Intersection",
"Set is Subset of Union",
"Definition:Topology"
] |
proofwiki-3844 | Excluded Set Topology is not T0 | Let $T = \struct {S, \tau_{\bar H} }$ be an excluded set space where $H$ has at least two distinct points.
Then $T$ is not a $T_0$ space. | Let $x, y \in H$ such that $x \ne y$.
Then $x, y \in S$, but by definition $x$ and $y$ are in no other open sets of $T$.
Hence there is no $U \in \tau_{\bar H}$ such that $x \in U, y \notin U$ or $y \in U, x \notin U$.
Hence the result, by definition of $T_0$ space.
{{qed}} | Let $T = \struct {S, \tau_{\bar H} }$ be an [[Definition:Excluded Set Topology|excluded set space]] where $H$ has at least two [[Definition:Distinct Elements|distinct points]].
Then $T$ is not a [[Definition:T0 Space|$T_0$ space]]. | Let $x, y \in H$ such that $x \ne y$.
Then $x, y \in S$, but by definition $x$ and $y$ are in no other [[Definition:Open Set (Topology)|open sets]] of $T$.
Hence there is no $U \in \tau_{\bar H}$ such that $x \in U, y \notin U$ or $y \in U, x \notin U$.
Hence the result, by definition of [[Definition:T0 Space|$T_0$ ... | Excluded Set Topology is not T0 | https://proofwiki.org/wiki/Excluded_Set_Topology_is_not_T0 | https://proofwiki.org/wiki/Excluded_Set_Topology_is_not_T0 | [
"Excluded Set Topologies",
"Examples of T0 Spaces"
] | [
"Definition:Excluded Set Topology",
"Definition:Distinct/Plural",
"Definition:T0 Space"
] | [
"Definition:Open Set/Topology",
"Definition:T0 Space"
] |
proofwiki-3845 | Either-Or Topology is Topology | Let $T = \struct {S, \tau}$ be the either-or topology.
Then $\tau$ is a topology on $T$. | {{Recall|Either-Or Topology|either-or topology}}
{{:Definition:Either-Or Topology}} | Let $T = \struct {S, \tau}$ be the [[Definition:Either-Or Topology|either-or topology]].
Then $\tau$ is a [[Definition:Topology|topology]] on $T$. | {{Recall|Either-Or Topology|either-or topology}}
{{:Definition:Either-Or Topology}} | Either-Or Topology is Topology | https://proofwiki.org/wiki/Either-Or_Topology_is_Topology | https://proofwiki.org/wiki/Either-Or_Topology_is_Topology | [
"Either-Or Topology"
] | [
"Definition:Either-Or Topology",
"Definition:Topology"
] | [
"Definition:Either-Or Topology",
"Definition:Either-Or Topology",
"Definition:Either-Or Topology",
"Definition:Either-Or Topology"
] |
proofwiki-3846 | Closed Sets of Either-Or Topology | Let $T = \struct {S, \tau}$ be the either-or space.
Then the closed sets of $T$ are:
:$\O$
:$S$
:$\set {-1}$
:$\set 1$
:$\set {-1, 1}$
:Every subset $H$ of $\closedint {-1} 1$ such that $\set 0 \subseteq H$. | From Open and Closed Sets in Topological Space we have that $\O$ and $S$ are closed sets trivially.
From the definition of closed set, we have:
:$U$ is open in $T$ {{iff}} $S \setminus U$ is closed in $T$
:$U$ is closed in $T$ {{iff}} $S \setminus U$ is open in $T$
where $S \setminus U$ denotes the relative complement ... | Let $T = \struct {S, \tau}$ be the [[Definition:Either-Or Space|either-or space]].
Then the [[Definition:Closed Set (Topology)|closed sets]] of $T$ are:
:$\O$
:$S$
:$\set {-1}$
:$\set 1$
:$\set {-1, 1}$
:Every [[Definition:Subset|subset]] $H$ of $\closedint {-1} 1$ such that $\set 0 \subseteq H$. | From [[Open and Closed Sets in Topological Space]] we have that $\O$ and $S$ are [[Definition:Closed Set (Topology)|closed sets]] trivially.
From the definition of [[Definition:Closed Set (Topology)|closed set]], we have:
:$U$ is [[Definition:Open Set (Topology)|open]] in $T$ {{iff}} $S \setminus U$ is [[Definition:C... | Closed Sets of Either-Or Topology | https://proofwiki.org/wiki/Closed_Sets_of_Either-Or_Topology | https://proofwiki.org/wiki/Closed_Sets_of_Either-Or_Topology | [
"Either-Or Topology"
] | [
"Definition:Either-Or Topology",
"Definition:Closed Set/Topology",
"Definition:Subset"
] | [
"Open and Closed Sets in Topological Space",
"Definition:Closed Set/Topology",
"Definition:Closed Set/Topology",
"Definition:Open Set/Topology",
"Definition:Closed Set/Topology",
"Definition:Closed Set/Topology",
"Definition:Open Set/Topology",
"Definition:Relative Complement",
"Definition:Open Set/... |
proofwiki-3847 | Boubaker's Theorem | Let $\struct {R, +, \circ}$ be a commutative ring.
Let $\struct {D, +, \circ}$ be an integral subdomain of $R$ whose zero is $0_D$ and whose unity is $1_D$.
Let $X \in R$ be transcendental over $D$.
Let $D \sqbrk X$ be the ring of polynomial forms in $X$ over $D$.
Finally, consider the following properties:
{{begin-eqn... | === Proof of validity ===
We first prove that the Boubaker Polynomials subsequence $\sequence {\map {B_{4 n} } x}$, defined in $D \sqbrk X$ verifies properties $(1)$, $(2)$, $(3)$ and $(4)$.
Let:
:$\struct {R, +, \circ}$ be a commutative ring
:$\struct {D, +, \circ}$ be an integral subdomain of $R$ whose zero is $0_D$ ... | Let $\struct {R, +, \circ}$ be a [[Definition:Commutative Ring|commutative ring]].
Let $\struct {D, +, \circ}$ be an [[Definition:Subdomain|integral subdomain]] of $R$ whose [[Definition:Ring Zero|zero]] is $0_D$ and whose [[Definition:Unity of Ring|unity]] is $1_D$.
Let $X \in R$ be [[Definition:Transcendental over ... | === Proof of validity ===
We first prove that the [[Definition:Boubaker Polynomials|Boubaker Polynomials]] [[Definition:Subsequence|subsequence]] $\sequence {\map {B_{4 n} } x}$, defined in $D \sqbrk X$ verifies properties $(1)$, $(2)$, $(3)$ and $(4)$.
Let:
:$\struct {R, +, \circ}$ be a [[Definition:Commutative Ring... | Boubaker's Theorem | https://proofwiki.org/wiki/Boubaker's_Theorem | https://proofwiki.org/wiki/Boubaker's_Theorem | [
"Boubaker Polynomials"
] | [
"Definition:Commutative Ring",
"Definition:Subdomain",
"Definition:Ring Zero",
"Definition:Unity (Abstract Algebra)/Ring",
"Definition:Transcendental (Abstract Algebra)/Ring",
"Definition:Ring of Polynomial Forms",
"Definition:Positive/Integer",
"Definition:Null Polynomial/Ring",
"Definition:Subsequ... | [
"Definition:Boubaker Polynomials",
"Definition:Subsequence",
"Definition:Commutative Ring",
"Definition:Subdomain",
"Definition:Ring Zero",
"Definition:Unity (Abstract Algebra)/Ring",
"Definition:Transcendental (Abstract Algebra)/Ring",
"Definition:Closed Form Expression",
"Definition:Boubaker Polyn... |
proofwiki-3848 | Boubaker's Theorem | Let $\struct {R, +, \circ}$ be a commutative ring.
Let $\struct {D, +, \circ}$ be an integral subdomain of $R$ whose zero is $0_D$ and whose unity is $1_D$.
Let $X \in R$ be transcendental over $D$.
Let $D \sqbrk X$ be the ring of polynomial forms in $X$ over $D$.
Finally, consider the following properties:
{{begin-eqn... | Let:
:$\struct {R, +, \circ}$ be a commutative ring
:$\struct {D, +, \circ}$ be an integral subdomain of $R$ whose zero is $0_D$ and whose unity is $1_D$
:$X \in R$ be transcendental over $D$.
It has been demonstrated that the Boubaker Polynomials sub-sequence $\map {B_{4 n} } x$, defined in $D \sqbrk X$ as:
:$\ds \map... | Let $\struct {R, +, \circ}$ be a [[Definition:Commutative Ring|commutative ring]].
Let $\struct {D, +, \circ}$ be an [[Definition:Subdomain|integral subdomain]] of $R$ whose [[Definition:Ring Zero|zero]] is $0_D$ and whose [[Definition:Unity of Ring|unity]] is $1_D$.
Let $X \in R$ be [[Definition:Transcendental over ... | Let:
:$\struct {R, +, \circ}$ be a [[Definition:Commutative Ring|commutative ring]]
:$\struct {D, +, \circ}$ be an [[Definition:Subdomain|integral subdomain]] of $R$ whose [[Definition:Ring Zero|zero]] is $0_D$ and whose [[Definition:Unity of Ring|unity]] is $1_D$
:$X \in R$ be [[Definition:Transcendental over Integral... | Boubaker's Theorem/Proof of Uniqueness | https://proofwiki.org/wiki/Boubaker's_Theorem | https://proofwiki.org/wiki/Boubaker's_Theorem/Proof_of_Uniqueness | [
"Boubaker Polynomials"
] | [
"Definition:Commutative Ring",
"Definition:Subdomain",
"Definition:Ring Zero",
"Definition:Unity (Abstract Algebra)/Ring",
"Definition:Transcendental (Abstract Algebra)/Ring",
"Definition:Ring of Polynomial Forms",
"Definition:Positive/Integer",
"Definition:Null Polynomial/Ring",
"Definition:Subsequ... | [
"Definition:Commutative Ring",
"Definition:Subdomain",
"Definition:Ring Zero",
"Definition:Unity (Abstract Algebra)/Ring",
"Definition:Transcendental (Abstract Algebra)/Ring",
"Definition:Boubaker Polynomials"
] |
proofwiki-3849 | Type Space is Compact | Let $\MM$ be an $\LL$-structure.
Let $A$ be a subset of the universe of $\MM$.
The type space $\map {S_n^\MM} A$ of $n$-types over $A$ is compact. | It will suffice to show that every open cover of $\map {S_n^\MM} A$ by the basic open sets $[\phi]$ of the topology has a finite subcover.
Let $\UU = \set { [\phi_i] : i \in I}$ be a cover of $\map {S_n^\MM} A$ by basic open sets.
This means that every complete $n$-type over $A$ contains some $\phi_i$.
We will find a f... | Let $\MM$ be an $\LL$-[[Definition:First-Order Structure|structure]].
Let $A$ be a subset of the universe of $\MM$.
The [[Definition:Type Space|type space]] $\map {S_n^\MM} A$ of $n$-types over $A$ is [[Definition:Compact Topological Space|compact]]. | It will suffice to show that every [[Definition:Open Cover|open cover]] of $\map {S_n^\MM} A$ by the [[Definition:Synthetic Basis|basic open sets]] $[\phi]$ of the topology has a finite [[Definition:Subcover|subcover]].
Let $\UU = \set { [\phi_i] : i \in I}$ be a cover of $\map {S_n^\MM} A$ by basic open sets.
This ... | Type Space is Compact | https://proofwiki.org/wiki/Type_Space_is_Compact | https://proofwiki.org/wiki/Type_Space_is_Compact | [
"Model Theory for Predicate Logic"
] | [
"Definition:Structure for Predicate Logic",
"Definition:Type Space",
"Definition:Compact Topological Space"
] | [
"Definition:Open Cover",
"Definition:Basis (Topology)/Synthetic Basis",
"Definition:Subcover",
"Definition:Subcover",
"Definition:Satisfiable",
"Definition:Type",
"Definition:Type",
"Compactness Theorem",
"Definition:Contradiction",
"Definition:Subcover",
"Category:Model Theory for Predicate Log... |
proofwiki-3850 | Either-Or Topology is T0 | Let $T = \struct {S, \tau}$ be the either-or space.
Then $T$ is a $T_0$ space. | Let $x, y \in S$ such that $x \ne y$.
{{WLOG}}, let $x \ne 0$.
Then $U = \set x$ is open in $T$ from the definition of the either-or topology.
We have that $x$ and $y$ are arbitrary.
So:
:$\forall x, y \in S: \exists U \in \tau: x \in U, y \notin U$
and the result follows by definition of $T_0$ space.
{{qed}} | Let $T = \struct {S, \tau}$ be the [[Definition:Either-Or Topology|either-or space]].
Then $T$ is a [[Definition:T0 Space|$T_0$ space]]. | Let $x, y \in S$ such that $x \ne y$.
{{WLOG}}, let $x \ne 0$.
Then $U = \set x$ is [[Definition:Open Set (Topology)|open]] in $T$ from the definition of the [[Definition:Either-Or Topology|either-or topology]].
We have that $x$ and $y$ are arbitrary.
So:
:$\forall x, y \in S: \exists U \in \tau: x \in U, y \notin ... | Either-Or Topology is T0 | https://proofwiki.org/wiki/Either-Or_Topology_is_T0 | https://proofwiki.org/wiki/Either-Or_Topology_is_T0 | [
"Either-Or Topology",
"Examples of T0 Spaces"
] | [
"Definition:Either-Or Topology",
"Definition:T0 Space"
] | [
"Definition:Open Set/Topology",
"Definition:Either-Or Topology",
"Definition:T0 Space"
] |
proofwiki-3851 | Either-Or Topology is not T1 | Let $T = \struct {S, \tau}$ be the either-or space.
Then $T$ is not a $T_1$ space. | Let $x = \dfrac 1 2$.
We have that $V = \set x$ such that $x \in V, 0 \notin V$.
However, by definition of the either-or topology, the only open sets of $T$ containing $0$ also contain $\openint {-1} 1$, and so must also contain $x$.
So we have that $\nexists U, V \in \tau: 0 \in U, x \notin U, x \in V, 0 \notin V$.
He... | Let $T = \struct {S, \tau}$ be the [[Definition:Either-Or Topology|either-or space]].
Then $T$ is not a [[Definition:T1 Space|$T_1$ space]]. | Let $x = \dfrac 1 2$.
We have that $V = \set x$ such that $x \in V, 0 \notin V$.
However, by definition of the [[Definition:Either-Or Topology|either-or topology]], the only [[Definition:Open Set (Topology)|open sets]] of $T$ containing $0$ also contain $\openint {-1} 1$, and so must also contain $x$.
So we have tha... | Either-Or Topology is not T1 | https://proofwiki.org/wiki/Either-Or_Topology_is_not_T1 | https://proofwiki.org/wiki/Either-Or_Topology_is_not_T1 | [
"Either-Or Topology",
"Examples of T1 Spaces"
] | [
"Definition:Either-Or Topology",
"Definition:T1 Space"
] | [
"Definition:Either-Or Topology",
"Definition:Open Set/Topology",
"Definition:T1 Space"
] |
proofwiki-3852 | Omitting Types Theorem | Let $\LL$ be the language of predicate logic with a countable signature.
Let $T$ be an $\LL$-theory.
Let $\set {p_i: i \in \N}$ be a countable set of non-isolated $n$-types of $T$.
There is a countable $\LL$-structure $\MM$ such that $\MM \models T$ and $\MM$ omits each $p_i$. | This proof is done by constructing a model using a method known as a Henkin construction.
This results in a model all of whose elements are the interpretations of constant symbols from some language.
The construction for this proof in particular is done so that the theory this model satisfies asserts that each tuple of... | Let $\LL$ be the [[Definition:Language of Predicate Logic|language of predicate logic]] with a [[Definition:Countable Set|countable]] [[Definition:Signature for Predicate Logic|signature]].
Let $T$ be an $\LL$-[[Definition:Theory|theory]].
Let $\set {p_i: i \in \N}$ be a [[Definition:Countable Set|countable set]] of ... | This proof is done by constructing a model using a method known as a Henkin construction.
This results in a model all of whose elements are the interpretations of constant symbols from some language.
The construction for this proof in particular is done so that the theory this model satisfies asserts that each tuple ... | Omitting Types Theorem | https://proofwiki.org/wiki/Omitting_Types_Theorem | https://proofwiki.org/wiki/Omitting_Types_Theorem | [
"Model Theory for Predicate Logic",
"Named Theorems"
] | [
"Definition:Language of Predicate Logic",
"Definition:Countable Set",
"Definition:Signature (Logic)/Predicate Logic",
"Definition:Theory",
"Definition:Countable Set",
"Definition:Isolated Type",
"Definition:Type",
"Definition:Countable Set",
"Definition:First Order Structure",
"Definition:Model (L... | [
"Maximal Finitely Satisfiable Theory with Witness Property is Satisfiable",
"Definition:Witness Property",
"Maximal Finitely Satisfiable Theory with Witness Property is Satisfiable",
"Maximal Finitely Satisfiable Theory with Witness Property is Satisfiable"
] |
proofwiki-3853 | Vinogradov's Theorem | Let $\Lambda$ be the von Mangoldt function.
For $N \in \Z$, let:
:$\ds \map R N = \sum_{n_1 + n_2 + n_3 \mathop = N} \map \Lambda {n_1} \, \map \Lambda {n_2} \, \map \Lambda {n_3}$
be a weighted count of the number of representations of $N$ as a sum of three prime powers.
Let $\SS$ be the arithmetic function:
:$\ds \ma... | Throughout the proof, for $\alpha \in \R$, let the following notation be understood:
:$\map e \alpha := \map \exp {2 \pi i \alpha}$
Let $B > 0$, and set $Q = \paren {\log N}^B$.
For $1 \le q \le Q, 0 \le a \le q$ such that $\gcd \set {a, q} = 1$, let:
:$\map \MM {q, a} := \set {\alpha \in \closedint 0 1: \size {\alpha ... | Let $\Lambda$ be the [[Definition:Von Mangoldt Function|von Mangoldt function]].
For $N \in \Z$, let:
:$\ds \map R N = \sum_{n_1 + n_2 + n_3 \mathop = N} \map \Lambda {n_1} \, \map \Lambda {n_2} \, \map \Lambda {n_3}$
be a [[Definition:Weighted Count|weighted count]] of the number of [[Definition:Representation|repr... | Throughout the proof, for $\alpha \in \R$, let the following notation be understood:
:$\map e \alpha := \map \exp {2 \pi i \alpha}$
Let $B > 0$, and set $Q = \paren {\log N}^B$.
For $1 \le q \le Q, 0 \le a \le q$ such that $\gcd \set {a, q} = 1$, let:
:$\map \MM {q, a} := \set {\alpha \in \closedint 0 1: \size {\alp... | Vinogradov's Theorem | https://proofwiki.org/wiki/Vinogradov's_Theorem | https://proofwiki.org/wiki/Vinogradov's_Theorem | [
"Vinogradov's Theorem",
"Analytic Number Theory",
"Number Theory"
] | [
"Definition:Von Mangoldt Function",
"Definition:Weighted Count",
"Definition:Representation",
"Definition:Prime Power",
"Definition:Arithmetic Function",
"Definition:Prime Number",
"Definition:Divisor (Algebra)/Integer",
"Definition:Divisor (Algebra)/Integer",
"Definition:Sufficiently Large",
"Def... | [] |
proofwiki-3854 | Vinogradov's Theorem/Minor Arcs | For any $B > 0$:
:$\ds \int_\MM \map F \alpha^3 \map e {-\alpha N} \rd \alpha \ll \frac {N^2} {\paren {\ln N}^{B/2 - 5} }$ | === Lemma 1 ===
{{:Vinogradov's Theorem/Minor Arcs/Lemma 1}}
{{qed|lemma}}
{{finish|The below need proofs}} | For any $B > 0$:
:$\ds \int_\MM \map F \alpha^3 \map e {-\alpha N} \rd \alpha \ll \frac {N^2} {\paren {\ln N}^{B/2 - 5} }$ | === [[Vinogradov's Theorem/Minor Arcs/Lemma 1|Lemma 1]] ===
{{:Vinogradov's Theorem/Minor Arcs/Lemma 1}}
{{qed|lemma}}
{{finish|The below need proofs}} | Vinogradov's Theorem/Minor Arcs | https://proofwiki.org/wiki/Vinogradov's_Theorem/Minor_Arcs | https://proofwiki.org/wiki/Vinogradov's_Theorem/Minor_Arcs | [
"Vinogradov's Theorem"
] | [] | [
"Vinogradov's Theorem/Minor Arcs/Lemma 1"
] |
proofwiki-3855 | Vinogradov's Theorem/Major Arcs | Let $B \in \R_{>0}$.
Then:
:$\ds \int_\MM \map F \alpha^3 \map e {-N \alpha} \rd \alpha = \frac {N^2} 2 \map \SS N + \map \OO {\frac {N^2} {\paren {\ln N}^{B/2} } }$
where the implied constant depends only on $B$. | === Lemma 1 ===
{{:Vinogradov's Theorem/Major Arcs/Lemma 1}}
{{qed|lemma}}
{{finish|These need proof.}} | Let $B \in \R_{>0}$.
Then:
:$\ds \int_\MM \map F \alpha^3 \map e {-N \alpha} \rd \alpha = \frac {N^2} 2 \map \SS N + \map \OO {\frac {N^2} {\paren {\ln N}^{B/2} } }$
where the [[Definition:Implied Constant|implied constant]] depends only on $B$. | === [[Vinogradov's Theorem/Major Arcs/Lemma 1|Lemma 1]] ===
{{:Vinogradov's Theorem/Major Arcs/Lemma 1}}
{{qed|lemma}}
{{finish|These need proof.}} | Vinogradov's Theorem/Major Arcs | https://proofwiki.org/wiki/Vinogradov's_Theorem/Major_Arcs | https://proofwiki.org/wiki/Vinogradov's_Theorem/Major_Arcs | [
"Vinogradov's Theorem"
] | [
"Definition:Big-O Notation/Implied Constant"
] | [
"Vinogradov's Theorem/Major Arcs/Lemma 1"
] |
proofwiki-3856 | Exponential Dominates Polynomial | Let $\exp$ denote the real exponential function.
Let $k \in \N$.
Let $\alpha \in \R_{>0}$.
Then:
:$\exists N \in \N: \forall x \in \R_{>N}: \map \exp {\alpha x} > x^k$ | Choose any $N > \dfrac {\paren {k + 1}!} {\alpha^{k + 1} }$, where $!$ denotes the factorial.
By Taylor Series Expansion for Exponential Function we have for $x \in \R_{\ge 0}$:
:$\ds \map \exp {\alpha x} = \sum_{m \mathop \ge 0} \frac {\paren {\alpha x}^m}{m!} > \frac {\paren {\alpha x}^{k + 1} } {\paren {k + 1}!}$
Th... | Let $\exp$ denote the [[Definition:Real Exponential Function|real exponential function]].
Let $k \in \N$.
Let $\alpha \in \R_{>0}$.
Then:
:$\exists N \in \N: \forall x \in \R_{>N}: \map \exp {\alpha x} > x^k$ | Choose any $N > \dfrac {\paren {k + 1}!} {\alpha^{k + 1} }$, where $!$ denotes the [[Definition:Factorial|factorial]].
By [[Taylor Series Expansion for Exponential Function]] we have for $x \in \R_{\ge 0}$:
:$\ds \map \exp {\alpha x} = \sum_{m \mathop \ge 0} \frac {\paren {\alpha x}^m}{m!} > \frac {\paren {\alpha x}^... | Exponential Dominates Polynomial | https://proofwiki.org/wiki/Exponential_Dominates_Polynomial | https://proofwiki.org/wiki/Exponential_Dominates_Polynomial | [
"Exponential Function",
"Polynomial Theory"
] | [
"Definition:Exponential Function/Real"
] | [
"Definition:Factorial",
"Power Series Expansion for Exponential Function",
"Category:Exponential Function",
"Category:Polynomial Theory"
] |
proofwiki-3857 | Power Dominates Logarithm | Let $\epsilon \in \R_{>0}$.
Let $B \in \N$ be arbitrary.
Then there exists $N \in \N$ such that:
:$\forall n > N: \paren {\ln n}^B < n^\epsilon$ | Let $\epsilon \in \R_{>0}$ be a (strictly) positive real number.
First we show that there exists $N$ such that $\ln n < n^\epsilon$ for all $n > N$,.
By Exponential is Strictly Increasing, the real exponential function is strictly increasing.
Therefore:
:$\ln n < n^\epsilon \iff n < \map \exp {n^\epsilon}$
Choose $k \i... | Let $\epsilon \in \R_{>0}$.
Let $B \in \N$ be arbitrary.
Then there exists $N \in \N$ such that:
:$\forall n > N: \paren {\ln n}^B < n^\epsilon$ | Let $\epsilon \in \R_{>0}$ be a [[Definition:Strictly Positive Real Number|(strictly) positive real number]].
First we show that there exists $N$ such that $\ln n < n^\epsilon$ for all $n > N$,.
By [[Exponential is Strictly Increasing]], the [[Definition:Real Exponential Function|real exponential function]] is [[Def... | Power Dominates Logarithm | https://proofwiki.org/wiki/Power_Dominates_Logarithm | https://proofwiki.org/wiki/Power_Dominates_Logarithm | [
"Logarithms",
"Powers"
] | [] | [
"Definition:Strictly Positive/Real Number",
"Exponential is Strictly Increasing",
"Definition:Exponential Function/Real",
"Definition:Strictly Increasing/Real Function",
"Power Series Expansion for Exponential Function",
"Category:Logarithms",
"Category:Powers"
] |
proofwiki-3858 | Cover of Interval By Closed Intervals is not Pairwise Disjoint | Let $\closedint a b$ be a closed interval in $\R$.
{{explain|Title mentions only "interval"; this does not affect truth of statement so may "closed" above line be removed as superfluous?}}
Let $\JJ$ be a set of two or more closed intervals contained in $\closedint a b$ such that $\ds \bigcup \JJ = \closedint a b$.
Then... | {{AimForCont}} that the intervals of $\JJ$ are pairwise disjoint.
Let $I = \closedint p q$ be the unique interval of $\JJ$ such that $a \in I$.
Let $J = \closedint r s$ be the unique interval of $\JJ$ containing the least real number not in $I$.
These choices are possible since $\JJ$ has at least two elements, and they... | Let $\closedint a b$ be a [[Definition:Closed Interval|closed interval]] in $\R$.
{{explain|Title mentions only "interval"; this does not affect truth of statement so may "closed" above line be removed as superfluous?}}
Let $\JJ$ be a [[Definition:Set|set]] of two or more [[Definition:Closed Interval|closed intervals... | {{AimForCont}} that the intervals of $\JJ$ are [[Definition:Pairwise Disjoint|pairwise disjoint]].
Let $I = \closedint p q$ be the [[Definition:Unique|unique]] [[Definition:Closed Interval|interval]] of $\JJ$ such that $a \in I$.
Let $J = \closedint r s$ be the [[Definition:Unique|unique]] [[Definition:Closed Interva... | Cover of Interval By Closed Intervals is not Pairwise Disjoint | https://proofwiki.org/wiki/Cover_of_Interval_By_Closed_Intervals_is_not_Pairwise_Disjoint | https://proofwiki.org/wiki/Cover_of_Interval_By_Closed_Intervals_is_not_Pairwise_Disjoint | [
"Measure Theory"
] | [
"Definition:Interval/Ordered Set/Closed",
"Definition:Set",
"Definition:Interval/Ordered Set/Closed",
"Definition:Pairwise Disjoint"
] | [
"Definition:Pairwise Disjoint",
"Definition:Unique",
"Definition:Interval/Ordered Set/Closed",
"Definition:Unique",
"Definition:Interval/Ordered Set/Closed",
"Definition:Real Number",
"Definition:Pairwise Disjoint",
"Definition:Real Number",
"Definition:Real Number",
"Definition:Pairwise Disjoint"... |
proofwiki-3859 | Open Extension Topology is T4 | Let $T = \struct {S, \tau}$ be a topological space.
Let $T^*_{\bar p} = \struct {S^*_p, \tau^*_{\bar p} }$ be the open extension space of $T$.
Then $T^*_{\bar p}$ is a $T_4$ space. | We have that an Open Extension Space is Ultraconnected.
That means none of its closed sets are disjoint.
Hence, vacuously, any two of its disjoint closed subsets are separated by neighborhoods.
The result follows by definition of $T_4$ space.
{{qed}} | Let $T = \struct {S, \tau}$ be a [[Definition:Topological Space|topological space]].
Let $T^*_{\bar p} = \struct {S^*_p, \tau^*_{\bar p} }$ be the [[Definition:Open Extension Space|open extension space]] of $T$.
Then $T^*_{\bar p}$ is a [[Definition:T4 Space|$T_4$ space]]. | We have that an [[Open Extension Space is Ultraconnected]].
That means none of its [[Definition:Closed Set (Topology)|closed sets]] are [[Definition:Disjoint Sets|disjoint]].
Hence, [[Definition:Vacuous Truth|vacuously]], any two of its [[Definition:Disjoint Sets|disjoint]] [[Definition:Closed Set (Topology)|closed s... | Open Extension Topology is T4 | https://proofwiki.org/wiki/Open_Extension_Topology_is_T4 | https://proofwiki.org/wiki/Open_Extension_Topology_is_T4 | [
"Open Extension Topologies",
"Examples of T4 Spaces"
] | [
"Definition:Topological Space",
"Definition:Open Extension Topology",
"Definition:T4 Space"
] | [
"Open Extension Space is Ultraconnected",
"Definition:Closed Set/Topology",
"Definition:Disjoint Sets",
"Definition:Vacuous Truth",
"Definition:Disjoint Sets",
"Definition:Closed Set/Topology",
"Definition:Separated by Neighborhoods",
"Definition:T4 Space"
] |
proofwiki-3860 | Excluded Point Topology is T4 | Let $T = \struct {S, \tau_{\bar p} }$ be an excluded point space.
Then $T$ is a $T_4$ space. | We have that an Excluded Point Space is Ultraconnected.
That means none of its closed sets are disjount.
Hence, vacuously, any two of its disjoint closed subsets of $S$ are separated by neighborhoods.
The result follows by definition of $T_4$ space.
{{qed}} | Let $T = \struct {S, \tau_{\bar p} }$ be an [[Definition:Excluded Point Topology|excluded point space]].
Then $T$ is a [[Definition:T4 Space|$T_4$ space]]. | We have that an [[Excluded Point Space is Ultraconnected]].
That means none of its [[Definition:Closed Set (Topology)|closed sets]] are [[Definition:Disjoint Sets|disjount]].
Hence, [[Definition:Vacuous Truth|vacuously]], any two of its [[Definition:Disjoint Sets|disjoint]] [[Definition:Closed Set (Topology)|closed s... | Excluded Point Topology is T4/Proof 1 | https://proofwiki.org/wiki/Excluded_Point_Topology_is_T4 | https://proofwiki.org/wiki/Excluded_Point_Topology_is_T4/Proof_1 | [
"Excluded Point Topology is T4",
"Excluded Point Topologies",
"Examples of T4 Spaces"
] | [
"Definition:Excluded Point Topology",
"Definition:T4 Space"
] | [
"Excluded Point Space is Ultraconnected",
"Definition:Closed Set/Topology",
"Definition:Disjoint Sets",
"Definition:Vacuous Truth",
"Definition:Disjoint Sets",
"Definition:Closed Set/Topology",
"Definition:Separated by Neighborhoods",
"Definition:T4 Space"
] |
proofwiki-3861 | Excluded Point Topology is T4 | Let $T = \struct {S, \tau_{\bar p} }$ be an excluded point space.
Then $T$ is a $T_4$ space. | We have:
: Excluded Point Topology is Open Extension Topology of Discrete Topology
: Open Extension Topology is $T_4$
Hence the result.
{{qed}} | Let $T = \struct {S, \tau_{\bar p} }$ be an [[Definition:Excluded Point Topology|excluded point space]].
Then $T$ is a [[Definition:T4 Space|$T_4$ space]]. | We have:
: [[Excluded Point Topology is Open Extension Topology of Discrete Topology]]
: [[Open Extension Topology is T4|Open Extension Topology is $T_4$]]
Hence the result.
{{qed}} | Excluded Point Topology is T4/Proof 2 | https://proofwiki.org/wiki/Excluded_Point_Topology_is_T4 | https://proofwiki.org/wiki/Excluded_Point_Topology_is_T4/Proof_2 | [
"Excluded Point Topology is T4",
"Excluded Point Topologies",
"Examples of T4 Spaces"
] | [
"Definition:Excluded Point Topology",
"Definition:T4 Space"
] | [
"Excluded Point Topology is Open Extension Topology of Discrete Topology",
"Open Extension Topology is T4"
] |
proofwiki-3862 | Excluded Point Topology is T4 | Let $T = \struct {S, \tau_{\bar p} }$ be an excluded point space.
Then $T$ is a $T_4$ space. | We have:
:Excluded Point Topology is $T_5$
:$T_5$ Space is $T_4$
{{qed}} | Let $T = \struct {S, \tau_{\bar p} }$ be an [[Definition:Excluded Point Topology|excluded point space]].
Then $T$ is a [[Definition:T4 Space|$T_4$ space]]. | We have:
:[[Excluded Point Topology is T5|Excluded Point Topology is $T_5$]]
:[[T5 Space is T4|$T_5$ Space is $T_4$]]
{{qed}} | Excluded Point Topology is T4/Proof 3 | https://proofwiki.org/wiki/Excluded_Point_Topology_is_T4 | https://proofwiki.org/wiki/Excluded_Point_Topology_is_T4/Proof_3 | [
"Excluded Point Topology is T4",
"Excluded Point Topologies",
"Examples of T4 Spaces"
] | [
"Definition:Excluded Point Topology",
"Definition:T4 Space"
] | [
"Excluded Point Space is T5",
"T5 Space is T4"
] |
proofwiki-3863 | Open Extension Topology is not T3 | Let $T = \struct {S, \tau}$ be a topological space.
Let $T^*_{\bar p} = \struct {S^*_p, \tau^*_{\bar p} }$ be the open extension space of $T$.
Then $T^*_{\bar p}$ is not a $T_3$ space. | As $S$ is (trivially) open in $T$, it is also open in $T^*_{\bar p}$.
As $S^*_p = S \cup \set p$, it follows that $\set p$ is closed in $T^*_{\bar p}$.
But by definition, the only open set of $T^*_{\bar p}$ that contains $\set p$ is $S^*_p$ itself.
So there can be no $x \in S^*_p: x \notin \set p$ contained in an open ... | Let $T = \struct {S, \tau}$ be a [[Definition:Topological Space|topological space]].
Let $T^*_{\bar p} = \struct {S^*_p, \tau^*_{\bar p} }$ be the [[Definition:Open Extension Space|open extension space]] of $T$.
Then $T^*_{\bar p}$ is not a [[Definition:T3 Space|$T_3$ space]]. | As $S$ is (trivially) [[Definition:Open Set (Topology)|open]] in $T$, it is also open in $T^*_{\bar p}$.
As $S^*_p = S \cup \set p$, it follows that $\set p$ is [[Definition:Closed Set (Topology)|closed]] in $T^*_{\bar p}$.
But by definition, the only [[Definition:Open Set (Topology)|open set]] of $T^*_{\bar p}$ that... | Open Extension Topology is not T3 | https://proofwiki.org/wiki/Open_Extension_Topology_is_not_T3 | https://proofwiki.org/wiki/Open_Extension_Topology_is_not_T3 | [
"Open Extension Topologies",
"Examples of T3 Spaces"
] | [
"Definition:Topological Space",
"Definition:Open Extension Topology",
"Definition:T3 Space"
] | [
"Definition:Open Set/Topology",
"Definition:Closed Set/Topology",
"Definition:Open Set/Topology",
"Definition:Open Set/Topology",
"Definition:Disjoint Sets"
] |
proofwiki-3864 | Excluded Point Topology is not T3 | Let $T = \struct {S, \tau_{\bar p} }$ be a excluded point space.
Then $T$ is not a $T_3$ space. | :Excluded Point Topology is Open Extension Topology of Discrete Topology
:Open Extension Topology is not T3
{{qed}} | Let $T = \struct {S, \tau_{\bar p} }$ be a [[Definition:Excluded Point Topology|excluded point space]].
Then $T$ is not a [[Definition:T3 Space|$T_3$ space]]. | :[[Excluded Point Topology is Open Extension Topology of Discrete Topology]]
:[[Open Extension Topology is not T3]]
{{qed}} | Excluded Point Topology is not T3 | https://proofwiki.org/wiki/Excluded_Point_Topology_is_not_T3 | https://proofwiki.org/wiki/Excluded_Point_Topology_is_not_T3 | [
"Excluded Point Topologies",
"Examples of T3 Spaces"
] | [
"Definition:Excluded Point Topology",
"Definition:T3 Space"
] | [
"Excluded Point Topology is Open Extension Topology of Discrete Topology",
"Open Extension Topology is not T3"
] |
proofwiki-3865 | Condition for Open Extension Space to be Separable | Let $T = \struct {S, \tau}$ be a topological space.
Let $T^*_{\bar p} = \struct {S^*_p, \tau^*_{\bar p} }$ be the open extension space of $T$.
Then $T^*_{\bar p}$ is a separable space {{iff}} $T$ is. | Let $T = \struct {S, \tau}$ be a separable space.
Then there exists a countable subset $H \subseteq S$ which is everywhere dense in $T$.
That is, $H^- = S$ where $H^-$ is the closure of $H$ in $S$.
So in $T_{\bar p}^*$, $S\subseteq H^-=S^-\subseteq S^*_p$.
Hence $H^- = S$ or $H^- = S \cup \set p = S^*_p$.
From Topologi... | Let $T = \struct {S, \tau}$ be a [[Definition:Topological Space|topological space]].
Let $T^*_{\bar p} = \struct {S^*_p, \tau^*_{\bar p} }$ be the [[Definition:Open Extension Space|open extension space]] of $T$.
Then $T^*_{\bar p}$ is a [[Definition:Separable Space|separable space]] {{iff}} $T$ is. | Let $T = \struct {S, \tau}$ be a [[Definition:Separable Space|separable space]].
Then there exists a [[Definition:Countable Set|countable]] [[Definition:Subset|subset]] $H \subseteq S$ which is [[Definition:Everywhere Dense|everywhere dense]] in $T$.
That is, $H^- = S$ where $H^-$ is the [[Definition:Closure (Topolo... | Condition for Open Extension Space to be Separable | https://proofwiki.org/wiki/Condition_for_Open_Extension_Space_to_be_Separable | https://proofwiki.org/wiki/Condition_for_Open_Extension_Space_to_be_Separable | [
"Open Extension Topologies",
"Examples of Separable Spaces"
] | [
"Definition:Topological Space",
"Definition:Open Extension Topology",
"Definition:Separable Space"
] | [
"Definition:Separable Space",
"Definition:Countable Set",
"Definition:Subset",
"Definition:Everywhere Dense",
"Definition:Closure (Topology)",
"Topological Closure is Closed",
"Definition:Closed Set/Topology",
"Definition:Open Extension Topology",
"Definition:Closed Set/Topology",
"Definition:Clos... |
proofwiki-3866 | Finitely Satisfiable Theory has Maximal Finitely Satisfiable Extension | Let $T$ be a finitely satisfiable $\LL$-theory.
Then there exists a finitely satisfiable $\LL$-theory $T'$ which contains $T$ as a subset such that:
:for all $\LL$-sentences $\phi$, either $\phi \in T'$ or $\neg \phi \in T'$. | The set of all finitely satisfiable $\LL$-theories containing $T$ forms an ordered set using subset inclusion as the ordering.
{{explain|Link to "subset is ordering"}}
Let $C$ be a non-empty chain in this ordered set.
Let $\ds T_C = \bigcup_{\Sigma \mathop \in C} \Sigma$.
Let $\Delta$ be a finite subset of $T_C$.
Then ... | Let $T$ be a [[Definition:Finitely Satisfiable|finitely satisfiable]] [[Definition:Theory|$\LL$-theory]].
Then there exists a [[Definition:Finitely Satisfiable|finitely satisfiable]] $\LL$-theory $T'$ which contains $T$ as a [[Definition:Subset|subset]] such that:
:for all $\LL$-sentences $\phi$, either $\phi \in T'$... | The [[Definition:Set|set]] of all [[Definition:Finitely Satisfiable|finitely satisfiable]] [[Definition:Theory|$\LL$-theories]] containing $T$ forms an [[Definition:Ordered Set|ordered set]] using [[Definition:Subset|subset inclusion]] as the [[Definition:Ordering|ordering]].
{{explain|Link to "subset is ordering"}}
... | Finitely Satisfiable Theory has Maximal Finitely Satisfiable Extension/Proof 1 | https://proofwiki.org/wiki/Finitely_Satisfiable_Theory_has_Maximal_Finitely_Satisfiable_Extension | https://proofwiki.org/wiki/Finitely_Satisfiable_Theory_has_Maximal_Finitely_Satisfiable_Extension/Proof_1 | [
"Model Theory for Predicate Logic"
] | [
"Definition:Finitely Satisfiable",
"Definition:Theory",
"Definition:Finitely Satisfiable",
"Definition:Subset"
] | [
"Definition:Set",
"Definition:Finitely Satisfiable",
"Definition:Theory",
"Definition:Ordered Set",
"Definition:Subset",
"Definition:Ordering",
"Definition:Non-Empty Set",
"Definition:Chain (Order Theory)/Subset Relation",
"Definition:Ordered Set",
"Definition:Finite Subset",
"Definition:Finitel... |
proofwiki-3867 | Finitely Satisfiable Theory has Maximal Finitely Satisfiable Extension | Let $T$ be a finitely satisfiable $\LL$-theory.
Then there exists a finitely satisfiable $\LL$-theory $T'$ which contains $T$ as a subset such that:
:for all $\LL$-sentences $\phi$, either $\phi \in T'$ or $\neg \phi \in T'$. | Let $\AA$ be the set of finitely satisfiable extensions of $T$.
By the lemma, for each element $S$ of $\AA$ and each $\LL$-sentence $\phi$, either $S \cup \set \phi \in \AA$ or $S \cup \set {\neg \phi} \in \AA$.
$\AA$ has finite character, by the following argument:
Let $S \in \AA$.
Let $F$ be a finite subset of $S$.
T... | Let $T$ be a [[Definition:Finitely Satisfiable|finitely satisfiable]] [[Definition:Theory|$\LL$-theory]].
Then there exists a [[Definition:Finitely Satisfiable|finitely satisfiable]] $\LL$-theory $T'$ which contains $T$ as a [[Definition:Subset|subset]] such that:
:for all $\LL$-sentences $\phi$, either $\phi \in T'$... | Let $\AA$ be the [[Definition:Set|set]] of finitely satisfiable extensions of $T$.
By the [[Finitely Satisfiable Theory has Maximal Finitely Satisfiable Extension/Lemma|lemma]], for each element $S$ of $\AA$ and each $\LL$-sentence $\phi$, either $S \cup \set \phi \in \AA$ or $S \cup \set {\neg \phi} \in \AA$.
$\AA$ ... | Finitely Satisfiable Theory has Maximal Finitely Satisfiable Extension/Proof 2 | https://proofwiki.org/wiki/Finitely_Satisfiable_Theory_has_Maximal_Finitely_Satisfiable_Extension | https://proofwiki.org/wiki/Finitely_Satisfiable_Theory_has_Maximal_Finitely_Satisfiable_Extension/Proof_2 | [
"Model Theory for Predicate Logic"
] | [
"Definition:Finitely Satisfiable",
"Definition:Theory",
"Definition:Finitely Satisfiable",
"Definition:Subset"
] | [
"Definition:Set",
"Finitely Satisfiable Theory has Maximal Finitely Satisfiable Extension/Lemma",
"Definition:Finite Character",
"Definition:Finite Subset",
"Definition:Finite Subset",
"Definition:Finite Subset",
"Restricted Tukey's Theorem"
] |
proofwiki-3868 | Condition for Open Extension Space to be First-Countable | Let $T = \struct {S, \tau}$ be a topological space.
Let $T^*_{\bar p} = \struct {S^*_p, \tau^*_{\bar p} }$ be the open extension space of $T$.
Then $T^*_{\bar p}$ is a first-countable space {{iff}} $T$ is. | Let $T = \struct {S, \tau}$ be a first-countable space.
Then every point in $S$ has a countable local basis.
Every open set of $T$ is an open set of $T^*_{\bar p}$ by definition.
So if $x$ has a countable local basis in $T$, then it has one in $T^*_{\bar p}$ as well.
Finally we note that $p$ is by definition in exactly... | Let $T = \struct {S, \tau}$ be a [[Definition:Topological Space|topological space]].
Let $T^*_{\bar p} = \struct {S^*_p, \tau^*_{\bar p} }$ be the [[Definition:Open Extension Space|open extension space]] of $T$.
Then $T^*_{\bar p}$ is a [[Definition:First-Countable Space|first-countable space]] {{iff}} $T$ is. | Let $T = \struct {S, \tau}$ be a [[Definition:First-Countable Space|first-countable space]].
Then every point in $S$ has a [[Definition:Countable Set|countable]] [[Definition:Local Basis|local basis]].
Every [[Definition:Open Set (Topology)|open set]] of $T$ is an [[Definition:Open Set (Topology)|open set]] of $T^*_{... | Condition for Open Extension Space to be First-Countable | https://proofwiki.org/wiki/Condition_for_Open_Extension_Space_to_be_First-Countable | https://proofwiki.org/wiki/Condition_for_Open_Extension_Space_to_be_First-Countable | [
"Open Extension Topologies",
"Examples of First-Countable Spaces"
] | [
"Definition:Topological Space",
"Definition:Open Extension Topology",
"Definition:First-Countable Space"
] | [
"Definition:First-Countable Space",
"Definition:Countable Set",
"Definition:Local Basis",
"Definition:Open Set/Topology",
"Definition:Open Set/Topology",
"Definition:Countable Set",
"Definition:Local Basis",
"Definition:Open Set/Topology",
"Definition:Countable Set",
"Definition:Local Basis",
"D... |
proofwiki-3869 | Condition for Open Extension Space to be Second-Countable | Let $T = \struct {S, \tau}$ be a topological space.
Let $T^*_{\bar p} = \struct {S^*_p, \tau^*_{\bar p} }$ be the open extension space of $T$.
Then $T^*_{\bar p}$ is a second-countable space {{iff}} $T$ is. | Let $T = \struct {S, \tau}$ be a second-countable space.
Then $\tau$ has a countable basis.
Every open set of $T$ is an open set of $T^*_{\bar p}$ by definition.
So if $\tau$ has a countable basis in $T$, then $\tau^*_{\bar p} = \tau \cup \set {S^*_p}$ has one as well.
So if $T$ is a second-countable space, then $T^*_{... | Let $T = \struct {S, \tau}$ be a [[Definition:Topological Space|topological space]].
Let $T^*_{\bar p} = \struct {S^*_p, \tau^*_{\bar p} }$ be the [[Definition:Open Extension Space|open extension space]] of $T$.
Then $T^*_{\bar p}$ is a [[Definition:Second-Countable Space|second-countable space]] {{iff}} $T$ is. | Let $T = \struct {S, \tau}$ be a [[Definition:Second-Countable Space|second-countable space]].
Then $\tau$ has a [[Definition:Countable Basis|countable basis]].
Every [[Definition:Open Set (Topology)|open set]] of $T$ is an [[Definition:Open Set (Topology)|open set]] of $T^*_{\bar p}$ by definition.
So if $\tau$ has... | Condition for Open Extension Space to be Second-Countable | https://proofwiki.org/wiki/Condition_for_Open_Extension_Space_to_be_Second-Countable | https://proofwiki.org/wiki/Condition_for_Open_Extension_Space_to_be_Second-Countable | [
"Open Extension Topologies",
"Examples of Second-Countable Spaces"
] | [
"Definition:Topological Space",
"Definition:Open Extension Topology",
"Definition:Second-Countable Space"
] | [
"Definition:Second-Countable Space",
"Definition:Countable Basis",
"Definition:Open Set/Topology",
"Definition:Open Set/Topology",
"Definition:Countable Basis",
"Definition:Second-Countable Space",
"Definition:Second-Countable Space",
"Definition:Second-Countable Space",
"Definition:Countable Basis"... |
proofwiki-3870 | Either-Or Topology is not T3 | Let $T = \struct {S, \tau}$ be the either-or space.
Then $T$ is not a $T_3$ space. | Consider the set $\set 0$ which is closed from Closed Sets of Either-Or Topology.
Let $x = \dfrac 1 2$.
Let $U \in \tau$ such that $\set 0 \subseteq U$.
Then by definition $\openint {-1} 1 \subseteq U$ and so $x \in U$.
So we have found a closed set of $T$ and a point in $S$ which do not satisfy the conditions for $T$ ... | Let $T = \struct {S, \tau}$ be the [[Definition:Either-Or Topology|either-or space]].
Then $T$ is not a [[Definition:T3 Space|$T_3$ space]]. | Consider the [[Definition:Set|set]] $\set 0$ which is [[Definition:Closed Set (Topology)|closed]] from [[Closed Sets of Either-Or Topology]].
Let $x = \dfrac 1 2$.
Let $U \in \tau$ such that $\set 0 \subseteq U$.
Then by definition $\openint {-1} 1 \subseteq U$ and so $x \in U$.
So we have found a [[Definition:Clos... | Either-Or Topology is not T3 | https://proofwiki.org/wiki/Either-Or_Topology_is_not_T3 | https://proofwiki.org/wiki/Either-Or_Topology_is_not_T3 | [
"Either-Or Topology",
"Examples of T3 Spaces"
] | [
"Definition:Either-Or Topology",
"Definition:T3 Space"
] | [
"Definition:Set",
"Definition:Closed Set/Topology",
"Closed Sets of Either-Or Topology",
"Definition:Closed Set/Topology",
"Definition:T3 Space"
] |
proofwiki-3871 | Limit Points of Either-Or Topology | Let $T = \struct {S, \tau}$ be the either-or space.
Let $H \subseteq S$ be any subset of $S$.
Then no element of $S$ can be a limit point of $H$ except $0$. | Let $x \in S$ such that $x \ne 0$.
Then, as $0 \notin \set x$, we have by definition of the either-or topology that $x$ is open in $T$.
So whatever $H$ is, $\set x$ never contains any points of $H$ which are different from $x$.
So $x$ can not be a limit point of $H$.
However, every open set of $T$ which contains $0$ al... | Let $T = \struct {S, \tau}$ be the [[Definition:Either-Or Space|either-or space]].
Let $H \subseteq S$ be any [[Definition:Subset|subset]] of $S$.
Then no element of $S$ can be a [[Definition:Limit Point of Set|limit point]] of $H$ except $0$. | Let $x \in S$ such that $x \ne 0$.
Then, as $0 \notin \set x$, we have by definition of the [[Definition:Either-Or Topology|either-or topology]] that $x$ is [[Definition:Open Set (Topology)|open]] in $T$.
So whatever $H$ is, $\set x$ never contains any [[Definition:Point of Set|points]] of $H$ which are different fro... | Limit Points of Either-Or Topology | https://proofwiki.org/wiki/Limit_Points_of_Either-Or_Topology | https://proofwiki.org/wiki/Limit_Points_of_Either-Or_Topology | [
"Examples of Limit Points",
"Either-Or Topology"
] | [
"Definition:Either-Or Topology",
"Definition:Subset",
"Definition:Limit Point/Topology/Set"
] | [
"Definition:Either-Or Topology",
"Definition:Open Set/Topology",
"Definition:Element",
"Definition:Limit Point/Topology/Set",
"Definition:Open Set/Topology",
"Definition:Real Interval/Open",
"Definition:Non-Empty Set",
"Definition:Limit Point/Topology/Set",
"Definition:Set Union"
] |
proofwiki-3872 | Either-Or Topology is T5 | Let $T = \struct {S, \tau}$ be the either-or space.
Then $T$ is a $T_5$ space. | Let $A, B \subseteq S$ such that $A \cap B \ne \O$.
Suppose neither $A$ nor $B$ contain $0$.
Then $A$ and $B$ are both open in $T$.
From Limit Points of Either-Or Topology, the only limit point that either $A$ or $B$ may have is $0$.
So either $A^- = A \cup \set 0$ or $A^- = A$, where $A^-$ is the closure of $A$
Simila... | Let $T = \struct {S, \tau}$ be the [[Definition:Either-Or Space|either-or space]].
Then $T$ is a [[Definition:T5 Space|$T_5$ space]]. | Let $A, B \subseteq S$ such that $A \cap B \ne \O$.
Suppose neither $A$ nor $B$ contain $0$.
Then $A$ and $B$ are both [[Definition:Open Set (Topology)|open]] in $T$.
From [[Limit Points of Either-Or Topology]], the only [[Definition:Limit Point of Set|limit point]] that either $A$ or $B$ may have is $0$.
So eithe... | Either-Or Topology is T5 | https://proofwiki.org/wiki/Either-Or_Topology_is_T5 | https://proofwiki.org/wiki/Either-Or_Topology_is_T5 | [
"Either-Or Topology",
"Examples of T5 Spaces"
] | [
"Definition:Either-Or Topology",
"Definition:T5 Space"
] | [
"Definition:Open Set/Topology",
"Limit Points of Either-Or Topology",
"Definition:Limit Point/Topology/Set",
"Definition:Closure (Topology)",
"Definition:Separated Sets",
"Definition:T5 Space",
"Definition:Separated Sets",
"Definition:Set Union",
"Definition:Disjoint Sets",
"Definition:Open Set/To... |
proofwiki-3873 | Either-Or Space is Lindelöf | Let $T = \struct {S, \tau}$ be the either-or space.
Then $T$ is a Lindelöf space. | We have:
:Either-Or Topology is Compact
:Compact Space is Lindelöf
Hence the result.
{{qed}} | Let $T = \struct {S, \tau}$ be the [[Definition:Either-Or Topology|either-or space]].
Then $T$ is a [[Definition:Lindelöf Space|Lindelöf space]]. | We have:
:[[Either-Or Topology is Compact]]
:[[Compact Space is Lindelöf]]
Hence the result.
{{qed}} | Either-Or Space is Lindelöf/Proof 1 | https://proofwiki.org/wiki/Either-Or_Space_is_Lindelöf | https://proofwiki.org/wiki/Either-Or_Space_is_Lindelöf/Proof_1 | [
"Either-Or Space is Lindelöf",
"Either-Or Topology",
"Examples of Lindelöf Spaces"
] | [
"Definition:Either-Or Topology",
"Definition:Lindelöf Space"
] | [
"Either-Or Topology is Compact",
"Compact Space is Lindelöf"
] |
proofwiki-3874 | Either-Or Space is Lindelöf | Let $T = \struct {S, \tau}$ be the either-or space.
Then $T$ is a Lindelöf space. | Any open cover $\CC$ of $T$ must contain an open set of $T$ which contains $0$.
So $\openint {-1} 1$ will always be covered by one set in $\CC$, leaving just $-1$ and $1$ possibly needing to be included in at most two other sets.
So $\CC$ has a subcover containing at most three sets.
Hence $T$ is a Lindelöf space by de... | Let $T = \struct {S, \tau}$ be the [[Definition:Either-Or Topology|either-or space]].
Then $T$ is a [[Definition:Lindelöf Space|Lindelöf space]]. | Any [[Definition:Open Cover|open cover]] $\CC$ of $T$ must contain an [[Definition:Open Set (Topology)|open set]] of $T$ which contains $0$.
So $\openint {-1} 1$ will always be [[Definition:Cover of Set|covered]] by one [[Definition:Set|set]] in $\CC$, leaving just $-1$ and $1$ possibly needing to be included in at mo... | Either-Or Space is Lindelöf/Proof 2 | https://proofwiki.org/wiki/Either-Or_Space_is_Lindelöf | https://proofwiki.org/wiki/Either-Or_Space_is_Lindelöf/Proof_2 | [
"Either-Or Space is Lindelöf",
"Either-Or Topology",
"Examples of Lindelöf Spaces"
] | [
"Definition:Either-Or Topology",
"Definition:Lindelöf Space"
] | [
"Definition:Open Cover",
"Definition:Open Set/Topology",
"Definition:Cover of Set",
"Definition:Set",
"Definition:Set",
"Definition:Subcover",
"Definition:Set",
"Definition:Lindelöf Space"
] |
proofwiki-3875 | Subspace of Either-Or Space less Zero is not Lindelöf | Let $T = \struct {S, \tau}$ be the either-or space.
Let $H = S \setminus \set 0$ be the set $S$ without zero.
Then the topological subspace $T_H = \struct {H, \tau_H}$ is not a Lindelöf space. | By definition of topological subspace, $U \subseteq H$ is open in $T_H$ {{iff}}:
:$(1): \quad \set 0 \nsubseteq U$
or:
:$(2): \quad \openint {-1} 1 \subseteq U$
But for all $U \subseteq H$, condition $(1)$ holds as $0 \notin H$.
So $T_H$ is by definition a discrete space.
As $T_H$ is uncountable, we have that Uncountab... | Let $T = \struct {S, \tau}$ be the [[Definition:Either-Or Topology|either-or space]].
Let $H = S \setminus \set 0$ be the [[Definition:Set|set]] $S$ without [[Definition:Zero (Number)|zero]].
Then the [[Definition:Topological Subspace|topological subspace]] $T_H = \struct {H, \tau_H}$ is not a [[Definition:Lindelöf ... | By definition of [[Definition:Topological Subspace|topological subspace]], $U \subseteq H$ is [[Definition:Open Set (Topology)|open]] in $T_H$ {{iff}}:
:$(1): \quad \set 0 \nsubseteq U$
or:
:$(2): \quad \openint {-1} 1 \subseteq U$
But for all $U \subseteq H$, condition $(1)$ holds as $0 \notin H$.
So $T_H$ is by def... | Subspace of Either-Or Space less Zero is not Lindelöf | https://proofwiki.org/wiki/Subspace_of_Either-Or_Space_less_Zero_is_not_Lindelöf | https://proofwiki.org/wiki/Subspace_of_Either-Or_Space_less_Zero_is_not_Lindelöf | [
"Lindelöf Spaces",
"Either-Or Topology"
] | [
"Definition:Either-Or Topology",
"Definition:Set",
"Definition:Zero (Number)",
"Definition:Topological Subspace",
"Definition:Lindelöf Space"
] | [
"Definition:Topological Subspace",
"Definition:Open Set/Topology",
"Definition:Discrete Topology",
"Definition:Uncountable/Set",
"Uncountable Discrete Space is not Lindelöf"
] |
proofwiki-3876 | Either-Or Topology is First-Countable | Let $T = \struct {S, \tau}$ be the either-or space.
Then $T$ is a first-countable space. | Let $x \in S$ such that $x \ne 0$.
Then $\set x$ is open in $T$ and so on its own forms a local basis of $x$ which is (trivially) countable.
Let $x = 0$.
Let $U \in \tau$ be open in $T$ such that $x \in U$.
Then by definition of the either-or space, $U$ contains the open set $\openint {-1} 1$.
So $\openint {-1} 1$ form... | Let $T = \struct {S, \tau}$ be the [[Definition:Either-Or Topology|either-or space]].
Then $T$ is a [[Definition:First-Countable Space|first-countable space]]. | Let $x \in S$ such that $x \ne 0$.
Then $\set x$ is [[Definition:Open Set (Topology)|open]] in $T$ and so on its own forms a [[Definition:Local Basis|local basis]] of $x$ which is (trivially) [[Definition:Countable Set|countable]].
Let $x = 0$.
Let $U \in \tau$ be [[Definition:Open Set (Topology)|open]] in $T$ such... | Either-Or Topology is First-Countable | https://proofwiki.org/wiki/Either-Or_Topology_is_First-Countable | https://proofwiki.org/wiki/Either-Or_Topology_is_First-Countable | [
"Either-Or Topology",
"Examples of First-Countable Spaces"
] | [
"Definition:Either-Or Topology",
"Definition:First-Countable Space"
] | [
"Definition:Open Set/Topology",
"Definition:Local Basis",
"Definition:Countable Set",
"Definition:Open Set/Topology",
"Definition:Either-Or Topology",
"Definition:Open Set/Topology",
"Definition:Local Basis",
"Definition:Countable Set",
"Definition:First-Countable Space"
] |
proofwiki-3877 | Either-Or Topology is not Separable | Let $T = \struct {S, \tau}$ be the either-or space.
Then $T$ is not a separable space. | From Limit Points of Either-Or Topology, the only limit point of any set of $S$ is $0$.
So the only set whose closure is $S$ are $S \setminus \set 0$ and $S$ itself.
So these two are the only subsets of $S$ which are everywhere dense in $S$.
Both of these are uncountable.
Hence the result, by definition of separable sp... | Let $T = \struct {S, \tau}$ be the [[Definition:Either-Or Topology|either-or space]].
Then $T$ is not a [[Definition:Separable Space|separable space]]. | From [[Limit Points of Either-Or Topology]], the only [[Definition:Limit Point of Set|limit point]] of any set of $S$ is $0$.
So the only set whose [[Definition:Closure (Topology)|closure]] is $S$ are $S \setminus \set 0$ and $S$ itself.
So these two are the only [[Definition:Subset|subsets]] of $S$ which are [[Defin... | Either-Or Topology is not Separable | https://proofwiki.org/wiki/Either-Or_Topology_is_not_Separable | https://proofwiki.org/wiki/Either-Or_Topology_is_not_Separable | [
"Either-Or Topology",
"Examples of Separable Spaces"
] | [
"Definition:Either-Or Topology",
"Definition:Separable Space"
] | [
"Limit Points of Either-Or Topology",
"Definition:Limit Point/Topology/Set",
"Definition:Closure (Topology)",
"Definition:Subset",
"Definition:Everywhere Dense",
"Definition:Uncountable/Set",
"Definition:Separable Space"
] |
proofwiki-3878 | Either-Or Topology is Non-Meager | Let $T = \struct {S, \tau}$ be the either-or space.
Then $T$ is non-meager. | From the definition of the either-or space, we have that every point $x$ in $T$ (apart from $0$) forms an open set of $T$.
So every non-empty subset of $T$ (apart from $\left\{{0}\right\})$ contains at least one open set of $T$.
So no subset of $T$ is nowhere dense in $T$.
So $T$ is not a countable union of subsets of ... | Let $T = \struct {S, \tau}$ be the [[Definition:Either-Or Topology|either-or space]].
Then $T$ is [[Definition:Non-Meager Space|non-meager]]. | From the definition of the [[Definition:Either-Or Topology|either-or space]], we have that every point $x$ in $T$ (apart from $0$) forms an [[Definition:Open Set (Topology)|open set]] of $T$.
So every [[Definition:Non-Empty Set|non-empty]] [[Definition:Subset|subset]] of $T$ (apart from $\left\{{0}\right\})$ contains ... | Either-Or Topology is Non-Meager/Proof 1 | https://proofwiki.org/wiki/Either-Or_Topology_is_Non-Meager | https://proofwiki.org/wiki/Either-Or_Topology_is_Non-Meager/Proof_1 | [
"Either-Or Topology is Non-Meager",
"Either-Or Topology",
"Examples of Non-Meager Spaces"
] | [
"Definition:Either-Or Topology",
"Definition:Meager Space/Non-Meager"
] | [
"Definition:Either-Or Topology",
"Definition:Open Set/Topology",
"Definition:Non-Empty Set",
"Definition:Subset",
"Definition:Open Set/Topology",
"Definition:Subset",
"Definition:Nowhere Dense",
"Definition:Set Union/Countable Union",
"Definition:Subset",
"Definition:Nowhere Dense",
"Definition:... |
proofwiki-3879 | Either-Or Topology is Non-Meager | Let $T = \struct {S, \tau}$ be the either-or space.
Then $T$ is non-meager. | From the definition of the either-or space, we have that every point $x$ in $T$ (apart from $0$) forms an open set of $T$.
The result follows directly from Space with Open Point is Non-Meager.
{{qed}} | Let $T = \struct {S, \tau}$ be the [[Definition:Either-Or Topology|either-or space]].
Then $T$ is [[Definition:Non-Meager Space|non-meager]]. | From the definition of the [[Definition:Either-Or Topology|either-or space]], we have that every point $x$ in $T$ (apart from $0$) forms an [[Definition:Open Set (Topology)|open set]] of $T$.
The result follows directly from [[Space with Open Point is Non-Meager]].
{{qed}} | Either-Or Topology is Non-Meager/Proof 2 | https://proofwiki.org/wiki/Either-Or_Topology_is_Non-Meager | https://proofwiki.org/wiki/Either-Or_Topology_is_Non-Meager/Proof_2 | [
"Either-Or Topology is Non-Meager",
"Either-Or Topology",
"Examples of Non-Meager Spaces"
] | [
"Definition:Either-Or Topology",
"Definition:Meager Space/Non-Meager"
] | [
"Definition:Either-Or Topology",
"Definition:Open Set/Topology",
"Space with Open Point is Non-Meager"
] |
proofwiki-3880 | Basis for Either-Or Topology | Let $T = \struct {S, \tau}$ be the either-or space.
Let $\BB$ be the set:
:$\BB := \set {\set x: x \in S, x \ne 0} \cup \set {\openint {-1} 1}$
that is, the set of all singleton subsets of $S$ less $\set 0$ and including the open real interval $\openint {-1} 1$.
Then $\BB$ is a basis for $T$. | Let $U \in \tau$ such that $0 \notin U$.
Then:
:$\ds U = \bigcup_{x \mathop \in U} \set x$
where $x \ne 0$.
Hence for all $x \in U$, we have $\set x \in \BB$.
Thus $U$ is the union of elements of $\BB$.
Now suppose $U \in \tau$ such that $0 \in U$.
Then $\openint {-1} 1 \subseteq U$ by definition.
So one of four cases ... | Let $T = \struct {S, \tau}$ be the [[Definition:Either-Or Topology|either-or space]].
Let $\BB$ be the [[Definition:Set|set]]:
:$\BB := \set {\set x: x \in S, x \ne 0} \cup \set {\openint {-1} 1}$
that is, the [[Definition:Set|set]] of all [[Definition:Singleton|singleton]] subsets of $S$ less $\set 0$ and includin... | Let $U \in \tau$ such that $0 \notin U$.
Then:
:$\ds U = \bigcup_{x \mathop \in U} \set x$
where $x \ne 0$.
Hence for all $x \in U$, we have $\set x \in \BB$.
Thus $U$ is the [[Definition:Set Union|union]] of [[Definition:Element|elements]] of $\BB$.
Now suppose $U \in \tau$ such that $0 \in U$.
Then $\openint ... | Basis for Either-Or Topology | https://proofwiki.org/wiki/Basis_for_Either-Or_Topology | https://proofwiki.org/wiki/Basis_for_Either-Or_Topology | [
"Either-Or Topology",
"Examples of Topological Bases"
] | [
"Definition:Either-Or Topology",
"Definition:Set",
"Definition:Set",
"Definition:Singleton",
"Definition:Real Interval/Open",
"Definition:Basis (Topology)"
] | [
"Definition:Set Union",
"Definition:Element",
"Definition:Set",
"Definition:Set Union",
"Definition:Element",
"Definition:Basis (Topology)",
"Category:Either-Or Topology",
"Category:Examples of Topological Bases"
] |
proofwiki-3881 | Either-Or Topology is Locally Path-Connected | Let $T = \struct {S, \tau}$ be the either-or space.
Then $T$ is a locally path-connected space. | Consider the set:
:$\BB := \set {\set x: x \in S, x \ne 0} \cup \openint {-1} 1$
Then by Basis for Either-Or Topology, $\BB$ is a basis for $T$.
From Point is Path-Connected to Itself, all $x \in S, x \ne 0$ are path-connected elements of $\BB$.
Finally we consider $\openint {-1} 1 \in \BB$.
Let $p \in \openint {-1} 1$... | Let $T = \struct {S, \tau}$ be the [[Definition:Either-Or Topology|either-or space]].
Then $T$ is a [[Definition:Locally Path-Connected Space|locally path-connected space]]. | Consider the set:
:$\BB := \set {\set x: x \in S, x \ne 0} \cup \openint {-1} 1$
Then by [[Basis for Either-Or Topology]], $\BB$ is a [[Definition:Basis (Topology)|basis]] for $T$.
From [[Point is Path-Connected to Itself]], all $x \in S, x \ne 0$ are [[Definition:Path-Connected Set|path-connected]] elements of $\BB$... | Either-Or Topology is Locally Path-Connected | https://proofwiki.org/wiki/Either-Or_Topology_is_Locally_Path-Connected | https://proofwiki.org/wiki/Either-Or_Topology_is_Locally_Path-Connected | [
"Either-Or Topology",
"Examples of Locally Path-Connected Spaces"
] | [
"Definition:Either-Or Topology",
"Definition:Locally Path-Connected Space"
] | [
"Basis for Either-Or Topology",
"Definition:Basis (Topology)",
"Point is Path-Connected to Itself",
"Definition:Path-Connected/Set",
"Definition:Mapping",
"Definition:Continuous Mapping (Topology)/Everywhere",
"Definition:Path (Topology)",
"Definition:Path-Connected/Set",
"Definition:Element",
"De... |
proofwiki-3882 | Either-Or Topology is Scattered | Let $T = \struct {S, \tau}$ be the either-or space.
Then $T$ is a scattered space. | {{Recall|Scattered Space|scattered space}}
{{:Definition:Scattered Space/Definition 1}}
Let $H$ be a non-empty subset of $S$.
Let $x \in H$.
Let $x \ne 0$.
By definition of either-or space, $\set x$ is open in $T$.
So $\set x$ is an open set of $x$ containing only $x$.
Thus by definition $x$ is isolated.
Thus if $H$ is... | Let $T = \struct {S, \tau}$ be the [[Definition:Either-Or Topology|either-or space]].
Then $T$ is a [[Definition:Scattered Space|scattered space]]. | {{Recall|Scattered Space|scattered space}}
{{:Definition:Scattered Space/Definition 1}}
Let $H$ be a [[Definition:Non-Empty Set|non-empty]] [[Definition:Subset|subset]] of $S$.
Let $x \in H$.
Let $x \ne 0$.
By definition of [[Definition:Either-Or Topology|either-or space]], $\set x$ is [[Definition:Open Set (Topolo... | Either-Or Topology is Scattered | https://proofwiki.org/wiki/Either-Or_Topology_is_Scattered | https://proofwiki.org/wiki/Either-Or_Topology_is_Scattered | [
"Either-Or Topology",
"Examples of Scattered Spaces"
] | [
"Definition:Either-Or Topology",
"Definition:Scattered Space"
] | [
"Definition:Non-Empty Set",
"Definition:Subset",
"Definition:Either-Or Topology",
"Definition:Open Set/Topology",
"Definition:Open Set/Topology",
"Definition:Isolated Point (Topology)",
"Definition:Non-Empty Set",
"Definition:Subset",
"Definition:Element",
"Definition:Isolated Point (Topology)",
... |
proofwiki-3883 | Countable Stability implies Stability for All Infinite Cardinalities | Let $T$ be a complete $\LL$-theory whose language $\LL$ is countable.
If $T$ is $\omega$-stable, then $T$ is $\kappa$-stable for all infinite $\kappa$. | We prove the contrapositive.
Let $\kappa$ be an infinite cardinal.
Suppose that $T$ is not $\kappa$-stable.
Then there exists some $\MM \models T$ and $A \subseteq \MM$ with $\card A = \kappa$ such that:
:$\card {\map { {S_n}^\MM} A} > \kappa$
Let $\LL_A$ denote $\LL \cup \set {a: a \in A}$, the language obtained from ... | Let $T$ be a [[Definition:Complete Theory|complete $\LL$-theory]] whose [[Definition:Logical Language|language]] $\LL$ is [[Definition:Countable Language|countable]].
If $T$ is [[Definition:Kappa-Stable Theory|$\omega$-stable]], then $T$ is [[Definition:Kappa-Stable Theory|$\kappa$-stable]] for all [[Definition:Infini... | We prove the [[Definition:Contrapositive Statement|contrapositive]].
Let $\kappa$ be an [[Definition:Infinite Cardinal|infinite cardinal]].
Suppose that $T$ is not [[Definition:Kappa-Stable Theory|$\kappa$-stable]].
Then there exists some $\MM \models T$ and $A \subseteq \MM$ with $\card A = \kappa$ such that:
:$\c... | Countable Stability implies Stability for All Infinite Cardinalities | https://proofwiki.org/wiki/Countable_Stability_implies_Stability_for_All_Infinite_Cardinalities | https://proofwiki.org/wiki/Countable_Stability_implies_Stability_for_All_Infinite_Cardinalities | [
"Model Theory for Predicate Logic",
"Countable Stability implies Stability for All Infinite Cardinalities"
] | [
"Definition:Complete Theory",
"Definition:Logical Language",
"Definition:Countable Language",
"Definition:Stability (Model Theory)/Kappa-Stable Theory",
"Definition:Stability (Model Theory)/Kappa-Stable Theory",
"Definition:Infinite Cardinal"
] | [
"Definition:Contrapositive Statement",
"Definition:Infinite Cardinal",
"Definition:Stability (Model Theory)/Kappa-Stable Theory",
"Definition:Logical Language",
"Definition:Constant Symbol",
"Definition:Well-Formed Formula",
"Definition:Set",
"Definition:Type",
"Definition:Countable Set",
"Definit... |
proofwiki-3884 | Finite Complement Topology is Topology | Let $T = \struct {S, \tau}$ be a finite complement space.
Then $\tau$ is a topology on $T$. | By definition, we have that $\O \in \tau$.
We also have that $S \in \tau$ as $\relcomp S S = \O$ which is trivially finite.
{{qed|lemma}}
Let $A, B \in \tau$.
Let $H = A \cap B$.
Then:
{{begin-eqn}}
{{eqn | l = H
| r = A \cap B
| c =
}}
{{eqn | ll= \leadsto
| l = \relcomp S H
| r = \relcomp S {A... | Let $T = \struct {S, \tau}$ be a [[Definition:Finite Complement Space|finite complement space]].
Then $\tau$ is a [[Definition:Topology|topology]] on $T$. | By definition, we have that $\O \in \tau$.
We also have that $S \in \tau$ as $\relcomp S S = \O$ which is trivially [[Definition:Finite Set|finite]].
{{qed|lemma}}
Let $A, B \in \tau$.
Let $H = A \cap B$.
Then:
{{begin-eqn}}
{{eqn | l = H
| r = A \cap B
| c =
}}
{{eqn | ll= \leadsto
| l = \relco... | Finite Complement Topology is Topology | https://proofwiki.org/wiki/Finite_Complement_Topology_is_Topology | https://proofwiki.org/wiki/Finite_Complement_Topology_is_Topology | [
"Finite Complement Topologies"
] | [
"Definition:Finite Complement Topology",
"Definition:Topology"
] | [
"Definition:Finite Set",
"De Morgan's Laws (Set Theory)/Relative Complement/Complement of Intersection",
"Definition:Finite Set",
"Definition:Set Union",
"Definition:Finite Set",
"Definition:Finite Set",
"Definition:Relative Complement",
"Definition:Finite Set",
"De Morgan's Laws (Set Theory)/Relati... |
proofwiki-3885 | Limit Points of Infinite Subset of Finite Complement Space | Let $T = \struct {S, \tau}$ be a finite complement space.
Let $H \subseteq S$ be an infinite subset of $S$.
Then every point of $S$ is a limit point of $H$. | Let $U \in \tau$ be any open set of $T$.
From Infinite Subset of Finite Complement Space Intersects Open Sets, we have that $U \cap H$ is infinite {{iff}} $H$ is infinite.
Let $x \in S$.
Then every open set $U$ in $T$ such that $x \in U$ also contains an infinite number of points of $H$ other than $x$.
Thus, by definit... | Let $T = \struct {S, \tau}$ be a [[Definition:Finite Complement Space|finite complement space]].
Let $H \subseteq S$ be an [[Definition:Infinite Set|infinite]] [[Definition:Subset|subset]] of $S$.
Then every point of $S$ is a [[Definition:Limit Point of Set|limit point]] of $H$. | Let $U \in \tau$ be any [[Definition:Open Set (Topology)|open set]] of $T$.
From [[Infinite Subset of Finite Complement Space Intersects Open Sets]], we have that $U \cap H$ is [[Definition:Infinite Set|infinite]] {{iff}} $H$ is [[Definition:Infinite Set|infinite]].
Let $x \in S$.
Then every [[Definition:Open Set (T... | Limit Points of Infinite Subset of Finite Complement Space | https://proofwiki.org/wiki/Limit_Points_of_Infinite_Subset_of_Finite_Complement_Space | https://proofwiki.org/wiki/Limit_Points_of_Infinite_Subset_of_Finite_Complement_Space | [
"Examples of Limit Points",
"Finite Complement Topologies"
] | [
"Definition:Finite Complement Topology",
"Definition:Infinite Set",
"Definition:Subset",
"Definition:Limit Point/Topology/Set"
] | [
"Definition:Open Set/Topology",
"Infinite Subset of Finite Complement Space Intersects Open Sets",
"Definition:Infinite Set",
"Definition:Infinite Set",
"Definition:Open Set/Topology",
"Definition:Infinite Set",
"Definition:Limit Point/Topology/Set"
] |
proofwiki-3886 | Finite Complement Topology is Separable | Let $T = \struct {S, \tau}$ be a finite complement topology on an infinite set $S$.
Then $T$ is a separable space. | Let $H$ be a countably infinite subset of $S$.
From Closure of Infinite Subset of Finite Complement Space, the closure of $H$ is $S$.
So by definition $H$ is everywhere dense in $T$.
Hence the result by definition of separable space.
{{qed}} | Let $T = \struct {S, \tau}$ be a [[Definition:Finite Complement Topology|finite complement topology]] on an [[Definition:Infinite Set|infinite]] set $S$.
Then $T$ is a [[Definition:Separable Space|separable space]]. | Let $H$ be a [[Definition:Countably Infinite Set|countably infinite subset]] of $S$.
From [[Closure of Infinite Subset of Finite Complement Space]], the [[Definition:Closure (Topology)|closure]] of $H$ is $S$.
So by definition $H$ is [[Definition:Everywhere Dense|everywhere dense]] in $T$.
Hence the result by defini... | Finite Complement Topology is Separable | https://proofwiki.org/wiki/Finite_Complement_Topology_is_Separable | https://proofwiki.org/wiki/Finite_Complement_Topology_is_Separable | [
"Finite Complement Topologies",
"Examples of Separable Spaces"
] | [
"Definition:Finite Complement Topology",
"Definition:Infinite Set",
"Definition:Separable Space"
] | [
"Definition:Countably Infinite/Set",
"Closure of Infinite Subset of Finite Complement Space",
"Definition:Closure (Topology)",
"Definition:Everywhere Dense",
"Definition:Separable Space"
] |
proofwiki-3887 | Closure of Infinite Subset of Finite Complement Space | Let $T = \struct {S, \tau}$ be a finite complement space.
Let $H \subseteq S$ be an infinite subset of $S$.
Then $H^- = S$ where $H^-$ is the closure of $S$. | Let $H$ be an infinite subset of $S$.
From Limit Points of Infinite Subset of Finite Complement Space, every point of $S$ is a limit point of $H$.
Hence the result from the definition of closure.
{{qed}} | Let $T = \struct {S, \tau}$ be a [[Definition:Finite Complement Space|finite complement space]].
Let $H \subseteq S$ be an [[Definition:Infinite Set|infinite]] [[Definition:Subset|subset]] of $S$.
Then $H^- = S$ where $H^-$ is the [[Definition:Closure (Topology)|closure]] of $S$. | Let $H$ be an [[Definition:Infinite Set|infinite]] [[Definition:Subset|subset]] of $S$.
From [[Limit Points of Infinite Subset of Finite Complement Space]], every point of $S$ is a [[Definition:Limit Point of Set|limit point]] of $H$.
Hence the result from the definition of [[Definition:Closure (Topology)|closure]].
... | Closure of Infinite Subset of Finite Complement Space | https://proofwiki.org/wiki/Closure_of_Infinite_Subset_of_Finite_Complement_Space | https://proofwiki.org/wiki/Closure_of_Infinite_Subset_of_Finite_Complement_Space | [
"Finite Complement Topologies",
"Examples of Set Closures"
] | [
"Definition:Finite Complement Topology",
"Definition:Infinite Set",
"Definition:Subset",
"Definition:Closure (Topology)"
] | [
"Definition:Infinite Set",
"Definition:Subset",
"Limit Points of Infinite Subset of Finite Complement Space",
"Definition:Limit Point/Topology/Set",
"Definition:Closure (Topology)"
] |
proofwiki-3888 | Subspace of Finite Complement Topology is Compact | Let $T = \struct {S, \tau}$ be a finite complement topology on an infinite set $S$.
Then every topological subspace of $T$, including $T$ itself, is a compact space. | Let $T_H = \struct {H, \tau_H}$ be a subspace of $T$.
Let $\CC$ be an open cover of $T_H$.
Let $U \in \CC$ be any set in $C$.
$U$ covers all but a finite number of points of $T_H$.
So for each of those points we pick an element of $\CC$ which covers each of those points.
Hence we have a finite subcover of $T_H$.
So by ... | Let $T = \struct {S, \tau}$ be a [[Definition:Finite Complement Topology|finite complement topology]] on an [[Definition:Infinite Set|infinite]] set $S$.
Then every [[Definition:Topological Subspace|topological subspace]] of $T$, including $T$ itself, is a [[Definition:Compact Topological Space|compact space]]. | Let $T_H = \struct {H, \tau_H}$ be a [[Definition:Topological Subspace|subspace]] of $T$.
Let $\CC$ be an [[Definition:Open Cover|open cover]] of $T_H$.
Let $U \in \CC$ be any [[Definition:Set|set]] in $C$.
$U$ [[Definition:Cover of Set|covers]] all but a [[Definition:Finite Set|finite number]] of points of $T_H$.
... | Subspace of Finite Complement Topology is Compact | https://proofwiki.org/wiki/Subspace_of_Finite_Complement_Topology_is_Compact | https://proofwiki.org/wiki/Subspace_of_Finite_Complement_Topology_is_Compact | [
"Finite Complement Topologies",
"Examples of Compact Topological Spaces"
] | [
"Definition:Finite Complement Topology",
"Definition:Infinite Set",
"Definition:Topological Subspace",
"Definition:Compact Topological Space"
] | [
"Definition:Topological Subspace",
"Definition:Open Cover",
"Definition:Set",
"Definition:Cover of Set",
"Definition:Finite Set",
"Definition:Element",
"Definition:Cover of Set",
"Definition:Subcover/Finite",
"Definition:Compact Topological Space"
] |
proofwiki-3889 | Uncountable Finite Complement Space is not Perfectly T4 | Let $T = \struct {S, \tau}$ be a finite complement space on an uncountable set $S$.
Then $T$ is not a perfectly $T_4$ space. | Recall the definition of a perfectly $T_4$ space
:Every closed set in $T$ can be written as a countable intersection of open sets of $T$.
Let $V$ be a closed set in $T$.
From Closed Set of Uncountable Finite Complement Topology is not $G_\delta$:
:$V$ is not a $G_\delta$ set.
The result follows by definition of perfect... | Let $T = \struct {S, \tau}$ be a [[Definition:Uncountable Finite Complement Space|finite complement space]] on an [[Definition:Uncountable Set|uncountable]] set $S$.
Then $T$ is not a [[Definition:Perfectly T4 Space|perfectly $T_4$ space]]. | Recall the definition of a [[Definition:Perfectly T4 Space|perfectly $T_4$ space]]
:Every [[Definition:Closed Set (Topology)|closed set]] in $T$ can be written as a [[Definition:Countable Intersection|countable intersection]] of [[Definition:Open Set (Topology)|open sets]] of $T$.
Let $V$ be a [[Definition:Closed Se... | Uncountable Finite Complement Space is not Perfectly T4 | https://proofwiki.org/wiki/Uncountable_Finite_Complement_Space_is_not_Perfectly_T4 | https://proofwiki.org/wiki/Uncountable_Finite_Complement_Space_is_not_Perfectly_T4 | [
"Uncountable Finite Complement Topologies",
"Examples of Perfectly T4 Spaces"
] | [
"Definition:Finite Complement Topology/Uncountable",
"Definition:Uncountable/Set",
"Definition:Perfectly T4 Space"
] | [
"Definition:Perfectly T4 Space",
"Definition:Closed Set/Topology",
"Definition:Set Intersection/Countable Intersection",
"Definition:Open Set/Topology",
"Definition:Closed Set/Topology",
"Closed Set of Uncountable Finite Complement Topology is not G-Delta",
"Definition:G-Delta Set",
"Definition:Perfec... |
proofwiki-3890 | Kluyver's Formula for Ramanujan's Sum | Let $q \in \N_{>0}$.
Let $n \in \N$.
Let $\map {c_q} n$ be Ramanujan's sum.
Let $\mu$ denote the Möbius function.
Then:
:$\ds \map {c_q} n = \sum_{d \mathop \divides \gcd \set {q, n} } d \map \mu {\frac q d}$
where $\divides$ denotes divisibility. | Let $\alpha \in \R$.
Let $e: \R \to \C$ be the mapping defined as:
:$\map e \alpha := \map \exp {2 \pi i \alpha}$
Let $\zeta_q$ be a primitive $q$th root of unity.
Let:
:$\ds \map {\eta_q} n := \sum_{1 \mathop \le a \mathop \le q} \map e {\frac {a n} q}$
By Complex Roots of Unity in Exponential Form this is the sum of ... | Let $q \in \N_{>0}$.
Let $n \in \N$.
Let $\map {c_q} n$ be [[Definition:Ramanujan Sum|Ramanujan's sum]].
Let $\mu$ denote the [[Definition:Möbius Function|Möbius function]].
Then:
:$\ds \map {c_q} n = \sum_{d \mathop \divides \gcd \set {q, n} } d \map \mu {\frac q d}$
where $\divides$ denotes [[Definition:Diviso... | Let $\alpha \in \R$.
Let $e: \R \to \C$ be the [[Definition:Mapping|mapping]] defined as:
:$\map e \alpha := \map \exp {2 \pi i \alpha}$
Let $\zeta_q$ be a [[Definition:Primitive Root of Unity|primitive $q$th root of unity]].
Let:
:$\ds \map {\eta_q} n := \sum_{1 \mathop \le a \mathop \le q} \map e {\frac {a n} q}... | Kluyver's Formula for Ramanujan's Sum | https://proofwiki.org/wiki/Kluyver's_Formula_for_Ramanujan's_Sum | https://proofwiki.org/wiki/Kluyver's_Formula_for_Ramanujan's_Sum | [
"Analytic Number Theory"
] | [
"Definition:Ramanujan Sum",
"Definition:Möbius Function",
"Definition:Divisor (Algebra)/Integer"
] | [
"Definition:Mapping",
"Definition:Root of Unity/Primitive",
"Complex Roots of Unity in Exponential Form",
"Definition:Root of Unity/Complex",
"Möbius Inversion Formula",
"Sum of Powers of Primitive Complex Roots of Unity",
"Common Divisor Divides GCD",
"Category:Analytic Number Theory"
] |
proofwiki-3891 | Condition for Complex Root of Unity to be Primitive | Let $n, k \in \N$.
Then $\alpha_k = \map \exp {\dfrac {2 \pi i k} n}$ is a primitive $n$th root of unity {{iff}} $\gcd \set {n, k} = 1$. | Let $U_n = \set {\map \exp {\dfrac {2 \pi i k} n}: 0 \le k \le n - 1}$.
Let $V = \set {1, \dotsc, {\alpha_k}^{n - 1} }$.
By Complex Roots of Unity in Exponential Form it is sufficient to show that $U_n = V$ {{iff}} $\gcd \set {n, k} = 1$.
Let $\gcd \set {n, k} = d > 1$.
Then there are $n', k' \in \N$ such that:
:$n' = ... | Let $n, k \in \N$.
Then $\alpha_k = \map \exp {\dfrac {2 \pi i k} n}$ is a [[Definition:Primitive Complex Root of Unity|primitive $n$th root of unity]] {{iff}} $\gcd \set {n, k} = 1$. | Let $U_n = \set {\map \exp {\dfrac {2 \pi i k} n}: 0 \le k \le n - 1}$.
Let $V = \set {1, \dotsc, {\alpha_k}^{n - 1} }$.
By [[Complex Roots of Unity in Exponential Form]] it is sufficient to show that $U_n = V$ {{iff}} $\gcd \set {n, k} = 1$.
Let $\gcd \set {n, k} = d > 1$.
Then there are $n', k' \in \N$ such that... | Condition for Complex Root of Unity to be Primitive | https://proofwiki.org/wiki/Condition_for_Complex_Root_of_Unity_to_be_Primitive | https://proofwiki.org/wiki/Condition_for_Complex_Root_of_Unity_to_be_Primitive | [
"Complex Roots of Unity"
] | [
"Definition:Root of Unity/Complex/Primitive"
] | [
"Complex Roots of Unity in Exponential Form",
"Definition:Distinct"
] |
proofwiki-3892 | Relation of Boubaker Polynomials to Chebyshev Polynomials | The Boubaker polynomials are related to Chebyshev polynomials through the equations:
:$\map {B_n} {2 x} = \dfrac {4 x} n \dfrac \d {\d x} \map {T_n} x - 2 \map {T_n} x$
:$\map {B_n} {2 x} = -2 \map {T_n} x + 4 x \map {U_{n - 1} } x$
where:
:$T_n$ denotes the Chebyshev polynomials of the first kind
:$U_n$ denotes the Ch... | ('''Using Riordan Matrix''')
The ordinary generating function of the Boubaker Polynomials can be expressed in terms of Riordan matrices:
:$\ds \sum_{n \geqslant 0} \map {B_n} t x^n = \frac {1 + 3 x^2} {1 - x t + x^2} = \paren {1 + 3 x^2 \mid 1 + x^2} \paren {\dfrac 1 {1 - x t} }$
By considering the Riordan matri... | The [[Definition:Boubaker Polynomials|Boubaker polynomials]] are related to [[Definition:Chebyshev Polynomials|Chebyshev polynomials]] through the equations:
:$\map {B_n} {2 x} = \dfrac {4 x} n \dfrac \d {\d x} \map {T_n} x - 2 \map {T_n} x$
:$\map {B_n} {2 x} = -2 \map {T_n} x + 4 x \map {U_{n - 1} } x$
where:
:$T_n$... | ('''Using Riordan Matrix''')
The ordinary generating function of the Boubaker Polynomials can be expressed in terms of [[Definition:Riordan Matrix|Riordan matrices]]:
:$\ds \sum_{n \geqslant 0} \map {B_n} t x^n = \frac {1 + 3 x^2} {1 - x t + x^2} = \paren {1 + 3 x^2 \mid 1 + x^2} \paren {\dfrac 1 {1 - x t} }$
... | Relation of Boubaker Polynomials to Chebyshev Polynomials | https://proofwiki.org/wiki/Relation_of_Boubaker_Polynomials_to_Chebyshev_Polynomials | https://proofwiki.org/wiki/Relation_of_Boubaker_Polynomials_to_Chebyshev_Polynomials | [
"Boubaker Polynomials",
"Chebyshev Polynomials"
] | [
"Definition:Boubaker Polynomials",
"Definition:Chebyshev Polynomials",
"Definition:Chebyshev Polynomials/First Kind",
"Definition:Chebyshev Polynomials/Second Kind"
] | [
"Definition:Riordan Matrix",
"Definition:Riordan Matrix",
"Definition:Chebyshev Polynomials",
"Definition:Chebyshev Polynomials/First Kind",
"Definition:Chebyshev Polynomials/Second Kind",
"Definition:Chebyshev Polynomials",
"Definition:Chebyshev Polynomials/First Kind",
"Definition:Chebyshev Polynomi... |
proofwiki-3893 | Relation of Boubaker Polynomials to Dickson Polynomials | The Boubaker polynomials $B_n$ are linked to the Dickson polynomials by the relations:
:$\map {B_{n + 1} } x \map {B_{n + j} } x - \map {B_{n + j + 1} } x \map {B_n} x = \paren {3 x^2 + 4} \map {D_{n + 1} } {x, \dfrac 1 4}$
:$\map {B_n} x = \map {D_n} {2 x, \dfrac 1 4} + 4 \map {D_{n - 1} } {2 x, \dfrac 1 4}$ | {{ProofWanted|The material on this page was originally raised by User:Yohji kinomoto.}} | The [[Definition:Boubaker Polynomials|Boubaker polynomials]] $B_n$ are linked to the [[Definition:Dickson Polynomials|Dickson polynomials]] by the relations:
:$\map {B_{n + 1} } x \map {B_{n + j} } x - \map {B_{n + j + 1} } x \map {B_n} x = \paren {3 x^2 + 4} \map {D_{n + 1} } {x, \dfrac 1 4}$
:$\map {B_n} x = \map ... | {{ProofWanted|The material on this page was originally raised by [[User:Yohji kinomoto]].}} | Relation of Boubaker Polynomials to Dickson Polynomials | https://proofwiki.org/wiki/Relation_of_Boubaker_Polynomials_to_Dickson_Polynomials | https://proofwiki.org/wiki/Relation_of_Boubaker_Polynomials_to_Dickson_Polynomials | [
"Boubaker Polynomials"
] | [
"Definition:Boubaker Polynomials",
"Definition:Dickson Polynomials"
] | [
"User:Yohji kinomoto"
] |
proofwiki-3894 | Relation of Boubaker Polynomials to Fermat Polynomials | The Boubaker polynomials are related to Fermat polynomials by:
:$\forall n \in \N: \map {B_n} x = \dfrac 1 {\paren {\sqrt 2}^n} \map {F_n} {\dfrac {2 \sqrt 2 x} 3} + \dfrac 3 {\paren {\sqrt 2}^{n - 2} } \map {F_{n - 2} } {\dfrac {2 \sqrt 2 x} 3}$ | ('''Using Riordan Matrix''')
Since the ordinary generating function of the Boubaker Polynomials can be expressed in terms of Riordan matrices:
:$\ds \sum_{n \mathop \geqslant 0}^{} \map {B_n} t x^n = \frac {1 + 3x^2 }{1 - x t + x^2} = \paren {1 + 3 x^2 \biggl\lvert 1 + x^2} \paren {\frac 1 {1 - x t} }$
then, by wr... | The [[Definition:Boubaker Polynomials|Boubaker polynomials]] are related to [[Definition:Fermat Polynomials|Fermat polynomials]] by:
:$\forall n \in \N: \map {B_n} x = \dfrac 1 {\paren {\sqrt 2}^n} \map {F_n} {\dfrac {2 \sqrt 2 x} 3} + \dfrac 3 {\paren {\sqrt 2}^{n - 2} } \map {F_{n - 2} } {\dfrac {2 \sqrt 2 x} 3}$ | ('''Using Riordan Matrix''')
Since the ordinary generating function of the Boubaker Polynomials can be expressed in terms of [[Definition:Riordan Matrix|Riordan matrices]]:
:$\ds \sum_{n \mathop \geqslant 0}^{} \map {B_n} t x^n = \frac {1 + 3x^2 }{1 - x t + x^2} = \paren {1 + 3 x^2 \biggl\lvert 1 + x^2} \paren {\frac ... | Relation of Boubaker Polynomials to Fermat Polynomials | https://proofwiki.org/wiki/Relation_of_Boubaker_Polynomials_to_Fermat_Polynomials | https://proofwiki.org/wiki/Relation_of_Boubaker_Polynomials_to_Fermat_Polynomials | [
"Boubaker Polynomials"
] | [
"Definition:Boubaker Polynomials",
"Definition:Fermat Polynomials"
] | [
"Definition:Riordan Matrix"
] |
proofwiki-3895 | F-Sigma and G-Delta Subsets of Uncountable Finite Complement Space | Let $T = \struct {S, \tau}$ be a finite complement topology on an uncountable set $S$.
Then countably infinite subsets of $S$ are $F_\sigma$ sets and are neither open nor closed sets.
Their relative complements in $S$ are $G_\delta$ sets, and are also neither open nor closed sets. | Let $U$ be a countably infinite subset of $S$.
Then it can be written as:
:$\ds U = \bigcup_{x \mathop \in U} \set x$
But as $\set x$ is a finite subset of $S$, it is by definition the complement of an open set in $T$.
So by definition $\set x$ is closed in $T$.
So:
:$\ds U = \bigcup_{x \mathop \in U} \set x$
is a coun... | Let $T = \struct {S, \tau}$ be a [[Definition:Uncountable Finite Complement Topology|finite complement topology]] on an [[Definition:Uncountable Set|uncountable]] set $S$.
Then [[Definition:Countably Infinite Set|countably infinite]] [[Definition:Subset|subsets]] of $S$ are [[Definition:F-Sigma Set|$F_\sigma$ sets]] ... | Let $U$ be a [[Definition:Countably Infinite Set|countably infinite]] subset of $S$.
Then it can be written as:
:$\ds U = \bigcup_{x \mathop \in U} \set x$
But as $\set x$ is a [[Definition:Finite Set|finite]] subset of $S$, it is by definition the [[Definition:Relative Complement|complement]] of an [[Definition:Open... | F-Sigma and G-Delta Subsets of Uncountable Finite Complement Space | https://proofwiki.org/wiki/F-Sigma_and_G-Delta_Subsets_of_Uncountable_Finite_Complement_Space | https://proofwiki.org/wiki/F-Sigma_and_G-Delta_Subsets_of_Uncountable_Finite_Complement_Space | [
"Uncountable Finite Complement Topologies",
"Examples of F-Sigma Sets",
"Examples of G-Delta Sets"
] | [
"Definition:Finite Complement Topology/Uncountable",
"Definition:Uncountable/Set",
"Definition:Countably Infinite/Set",
"Definition:Subset",
"Definition:F-Sigma Set",
"Definition:Open Set/Topology",
"Definition:Closed Set/Topology",
"Definition:Relative Complement",
"Definition:G-Delta Set",
"Defi... | [
"Definition:Countably Infinite/Set",
"Definition:Finite Set",
"Definition:Relative Complement",
"Definition:Open Set/Topology",
"Definition:Closed Set/Topology",
"Definition:Set Union/Countable Union",
"Definition:Closed Set/Topology",
"Definition:F-Sigma Set",
"Definition:Countably Infinite/Set",
... |
proofwiki-3896 | Skolem's Paradox | Let $\LL$ be a countable first-order language.
Let $T$ be an $\LL$-theory which axiomatizes some version of set theory (for example, ZFC).
There is a countable model of $T$. | This is a straightforward application of the downward Löwenheim-Skolem Theorem.
{{qed}}
{{refactor|From here on down, decide where it belongs and how it is to be presented. At the moment it is too much like an encyclopedia article to be compatible with {{ProofWiki}}'s dictionary style.|level = advanced}} | Let $\LL$ be a [[Definition:Countable|countable]] [[Definition:First Order Logic|first-order]] [[Definition:Logical Language|language]].
Let $T$ be an $\LL$-[[Definition:Theory|theory]] which [[Definition:Axiomatization|axiomatizes]] some version of [[Definition:Axiomatic Set Theory|set theory]] (for example, [[Defini... | This is a straightforward application of the downward [[Löwenheim-Skolem Theorem]].
{{qed}}
{{refactor|From here on down, decide where it belongs and how it is to be presented. At the moment it is too much like an encyclopedia article to be compatible with {{ProofWiki}}'s dictionary style.|level = advanced}} | Skolem's Paradox | https://proofwiki.org/wiki/Skolem's_Paradox | https://proofwiki.org/wiki/Skolem's_Paradox | [
"Mathematical Logic",
"Set Theory"
] | [
"Definition:Countable Set",
"Definition:Predicate Logic",
"Definition:Logical Language",
"Definition:Theory",
"Definition:Axiomatization",
"Definition:Axiomatic Set Theory",
"Definition:Zermelo-Fraenkel Set Theory with Axiom of Choice"
] | [
"Löwenheim-Skolem Theorem"
] |
proofwiki-3897 | Uncountable Finite Complement Space is not First-Countable | Let $T = \struct {S, \tau}$ be a finite complement topology on an uncountable set $S$.
Then $T$ is not first-countable. | {{AimForCont}} some $x \in S$ has a countable local basis.
That means:
:there exists a countable set $\BB_x \subseteq \tau$
such that:
:$\forall B \in \BB_x: x \in B$
and such that:
:every open neighborhood of $x$ contains some $B \in \BB_x$.
Let $y \in S$ with $y \ne x$.
Because its complement relative to $S$ is finit... | Let $T = \struct {S, \tau}$ be a [[Definition:Uncountable Finite Complement Topology|finite complement topology]] on an [[Definition:Uncountable Set|uncountable set]] $S$.
Then $T$ is not [[Definition:First-Countable Space|first-countable]]. | {{AimForCont}} some $x \in S$ has a [[Definition:Countable Set|countable]] [[Definition:Local Basis|local basis]].
That means:
:there exists a [[Definition:Countable Set|countable]] [[Definition:Set of Sets|set]] $\BB_x \subseteq \tau$
such that:
:$\forall B \in \BB_x: x \in B$
and such that:
:every [[Definition:Open... | Uncountable Finite Complement Space is not First-Countable | https://proofwiki.org/wiki/Uncountable_Finite_Complement_Space_is_not_First-Countable | https://proofwiki.org/wiki/Uncountable_Finite_Complement_Space_is_not_First-Countable | [
"Uncountable Finite Complement Topologies",
"Examples of First-Countable Spaces"
] | [
"Definition:Finite Complement Topology/Uncountable",
"Definition:Uncountable/Set",
"Definition:First-Countable Space"
] | [
"Definition:Countable Set",
"Definition:Local Basis",
"Definition:Countable Set",
"Definition:Set of Sets",
"Definition:Open Neighborhood/Point",
"Definition:Relative Complement",
"Definition:Finite Set",
"Definition:Open Set/Topology",
"Definition:Neighborhood Basis",
"De Morgan's Laws (Set Theor... |
proofwiki-3898 | Separable Space is not necessarily First-Countable | Let $T = \struct {S, \tau}$ be a topological space which is separable.
Then it is not necessarily the case that $T$ is first-countable. | ;Proof by Counterexample
Let $T = \struct {S, \tau}$ be a finite complement topology on an uncountable set $S$.
We have that a Finite Complement Topology is Separable.
But we also have that an Uncountable Finite Complement Space is not First-Countable.
Hence the result.
{{qed}} | Let $T = \struct {S, \tau}$ be a [[Definition:Topological Space|topological space]] which is [[Definition:Separable Space|separable]].
Then it is not necessarily the case that $T$ is [[Definition:First-Countable Space|first-countable]]. | ;[[Proof by Counterexample]]
Let $T = \struct {S, \tau}$ be a [[Definition:Uncountable Finite Complement Topology|finite complement topology]] on an [[Definition:Uncountable Set|uncountable]] set $S$.
We have that a [[Finite Complement Topology is Separable]].
But we also have that an [[Uncountable Finite Complement... | Separable Space is not necessarily First-Countable | https://proofwiki.org/wiki/Separable_Space_is_not_necessarily_First-Countable | https://proofwiki.org/wiki/Separable_Space_is_not_necessarily_First-Countable | [
"First-Countable Spaces",
"Separable Spaces"
] | [
"Definition:Topological Space",
"Definition:Separable Space",
"Definition:First-Countable Space"
] | [
"Proof by Counterexample",
"Definition:Finite Complement Topology/Uncountable",
"Definition:Uncountable/Set",
"Finite Complement Topology is Separable",
"Uncountable Finite Complement Space is not First-Countable"
] |
proofwiki-3899 | Ramanujan Sum is Multiplicative | Let $q \in \N_{>0}$, $n \in \N$.
Let $\map {c_q} n$ be the Ramanujan sum.
Then $\map {c_q} n$ is multiplicative in $q$. | Let $q, r \in \N$ such that:
:$\gcd \set {q, r} = 1$
where $\gcd \set {q, r}$ denotes the greatest common divisor of $q$ and $r$.
Then:
{{begin-eqn}}
{{eqn | l = \map {c_q} n \, \map {c_r} n
| r = \sum_{d_1 \mathop \divides \gcd \set {n, q} } d_1 \, \map \mu {\frac q {d_1} } \sum_{d_2 \mathop \divides \gcd \set {... | Let $q \in \N_{>0}$, $n \in \N$.
Let $\map {c_q} n$ be the [[Definition:Ramanujan Sum|Ramanujan sum]].
Then $\map {c_q} n$ is [[Definition:Multiplicative Arithmetic Function|multiplicative]] in $q$. | Let $q, r \in \N$ such that:
:$\gcd \set {q, r} = 1$
where $\gcd \set {q, r}$ denotes the [[Definition:Greatest Common Divisor of Integers|greatest common divisor]] of $q$ and $r$.
Then:
{{begin-eqn}}
{{eqn | l = \map {c_q} n \, \map {c_r} n
| r = \sum_{d_1 \mathop \divides \gcd \set {n, q} } d_1 \, \map \mu {\f... | Ramanujan Sum is Multiplicative | https://proofwiki.org/wiki/Ramanujan_Sum_is_Multiplicative | https://proofwiki.org/wiki/Ramanujan_Sum_is_Multiplicative | [
"Analytic Number Theory"
] | [
"Definition:Ramanujan Sum",
"Definition:Multiplicative Arithmetic Function"
] | [
"Definition:Greatest Common Divisor/Integers",
"Kluyver's Formula for Ramanujan's Sum",
"Möbius Function is Multiplicative",
"GCD with One Fixed Argument is Multiplicative Function",
"Kluyver's Formula for Ramanujan's Sum",
"Category:Analytic Number Theory"
] |
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