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nLab > Latest Changes: countably compact topological space

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- CommentRowNumber1.
- CommentAuthorTodd_Trimble
- CommentTimeJul 25th 2018
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Author: Todd_Trimble
Format: MarkdownItexI added the example of the [[long line]]. <a href="https://ncatlab.org/nlab/revision/diff/countably+compact+topological+space/4">diff</a>, <a href="https://ncatlab.org/nlab/revision/countably+compact+topological+space/4">v4</a>, <a href="https://ncatlab.org/nlab/show/countably+compact+topological+space">current</a>

I added the example of the long line.

diff, v4, current
- CommentRowNumber2.
- CommentAuthorTodd_Trimble
- CommentTimeJul 27th 2018
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Author: Todd_Trimble
Format: MarkdownItexAdded some results indicating the relation between countable compactness and limit point compactness. <a href="https://ncatlab.org/nlab/revision/diff/countably+compact+topological+space/5">diff</a>, <a href="https://ncatlab.org/nlab/revision/countably+compact+topological+space/5">v5</a>, <a href="https://ncatlab.org/nlab/show/countably+compact+topological+space">current</a>

Added some results indicating the relation between countable compactness and limit point compactness.

diff, v5, current
- CommentRowNumber3.
- CommentAuthorTodd_Trimble
- CommentTimeJul 27th 2018
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Author: Todd_Trimble
Format: MarkdownItexDespite the announcement of #2, the changes I made are not being displayed. I'm not seeing any syntax errors.

Despite the announcement of #2, the changes I made are not being displayed. I’m not seeing any syntax errors.
- CommentRowNumber4.
- CommentAuthorTodd_Trimble
- CommentTimeJul 27th 2018
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Author: Todd_Trimble
Format: MarkdownItexI'm recording my edit here, to be on the safe side. * An uncountable set equipped with the cocountable topology is countably compact. ## Properties +-- {: .num_prop} ###### Proposition A countably compact space is a [[limit compact space]]. =-- +-- {: .proof} ###### Proof Recall that a space is limit point compact if every [[closed subspace|closed]] [[discrete space|discrete]] [[subspace]] is [[finite set|finite]]; equivalently, if every [[countable set|countable]] closed discrete subspace is finite. Suppose $X$ is countably compact and $A$ is a countable closed discrete subspace. For each $a \in A$, choose an open neighborhood $U_a$ such that $U_a \cap A = \{a\}$, and let $V_a$ be the open subset $U_a \cup \neg A$. Clearly we still have $V_a \cap A = \{a\}$. Also, $\{V_a: a \in A\}$ is a countable cover of $X$, hence admits a finite subcover $V_{a_1}, \ldots, V_{a_n}$. But then $$A = \left(\bigcup_{i=1}^n V_{a_i} \right) \cap A = \bigcup_{i=1}^n (V_{a_i} \cap A) = \{a_1, \ldots, a_n\}$$ as was to be shown. =-- +-- {: .num_prop} ###### Proposition A space that is $T_1$ and limit point compact is countably compact. =-- +-- {: .proof} ###### Proof Equivalently (under classical logic), the assertion is that if a $T_1$ space $X$ is not countably compact, then it is not limit point compact. Indeed, under this hypothesis, there is a countable open cover $U_1, U_2, \ldots$ of $X$ that admits no finite subcover. We put $V_n = U_1 \cup \ldots \cup U_n$ so that $V_1 \subseteq V_2 \subseteq \ldots$; since every point belongs to some $V_n$ but no $V_n$ is all of $X$, we may discard repetitions and assume without loss of generality that all the inclusions $$\emptyset = V_0 \subset V_1 \subset V_2 \subset \ldots$$ are strict. Thus for each $n \geq 1$ we may pick a point $x_n \in V_n \setminus V_{n-1}$. Observe that if $m \lt n$, then $x_n \notin V_m$. Since $X$ is $T_1$ (points are closed), the set $W_m = V_m \cap \neg \{x_1, \ldots, x_{m-1}\}$ is an open neighborhood of $x_m$ that does not contain $x_n$ whenever $m \lt n$, and does not contain $x_n$ for $n \lt m$. Thus every point $x_n$ is open relative to $A = \{x_1, x_2, \ldots\}$, i.e., $A$ is a discrete subspace. Finally, any point $x \notin A$ belongs to some $V_n$, and then $V_n \cap \neg \{x_1, \ldots x_n\}$ is an open neighborhood of $x$ that doesn't intersect $A$. Thus $A$ is an infinite closed discrete subspace, meaning that $X$ is not limit point compact. =-- * In the case of metric spaces, [[countably compact metric spaces are equivalently compact metric spaces]].
I’m recording my edit here, to be on the safe side.
- An uncountable set equipped with the cocountable topology is countably compact.
Properties

+– {: .num_prop}

Proposition

A countably compact space is a limit compact space. =–

+– {: .proof}

Proof

Recall that a space is limit point compact if every closed discrete subspace is finite; equivalently, if every countable closed discrete subspace is finite.

Suppose $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>$ is countably compact and $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math>$ is a countable closed discrete subspace. For each $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>a</mi><mo>∈</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">a \in A</annotation></semantics></math>$ , choose an open neighborhood $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>U</mi> <mi>a</mi></msub></mrow><annotation encoding="application/x-tex">U_a</annotation></semantics></math>$ such that $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>U</mi> <mi>a</mi></msub><mo>∩</mo><mi>A</mi><mo>=</mo><mo stretchy="false">{</mo><mi>a</mi><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">U_a \cap A = \{a\}</annotation></semantics></math>$ , and let $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>V</mi> <mi>a</mi></msub></mrow><annotation encoding="application/x-tex">V_a</annotation></semantics></math>$ be the open subset $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>U</mi> <mi>a</mi></msub><mo>∪</mo><mo>¬</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">U_a \cup \neg A</annotation></semantics></math>$ . Clearly we still have $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>V</mi> <mi>a</mi></msub><mo>∩</mo><mi>A</mi><mo>=</mo><mo stretchy="false">{</mo><mi>a</mi><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">V_a \cap A = \{a\}</annotation></semantics></math>$ . Also, $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo stretchy="false">{</mo><msub><mi>V</mi> <mi>a</mi></msub><mo>:</mo><mi>a</mi><mo>∈</mo><mi>A</mi><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{V_a: a \in A\}</annotation></semantics></math>$ is a countable cover of $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>$ , hence admits a finite subcover $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>V</mi> <mrow><msub><mi>a</mi> <mn>1</mn></msub></mrow></msub><mo>,</mo><mi>…</mi><mo>,</mo><msub><mi>V</mi> <mrow><msub><mi>a</mi> <mi>n</mi></msub></mrow></msub></mrow><annotation encoding="application/x-tex">V_{a_1}, \ldots, V_{a_n}</annotation></semantics></math>$ . But then
A=(n⋃i=1Vai)∩A=n⋃i=1(Vai∩A)={a1,…,an}

as was to be shown.
=–

+– {: .num_prop}

Proposition

A space that is $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>T</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">T_1</annotation></semantics></math>$ and limit point compact is countably compact. =–

+– {: .proof}

Proof

Equivalently (under classical logic), the assertion is that if a $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>T</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">T_1</annotation></semantics></math>$ space $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>$ is not countably compact, then it is not limit point compact. Indeed, under this hypothesis, there is a countable open cover $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>U</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>U</mi> <mn>2</mn></msub><mo>,</mo><mi>…</mi></mrow><annotation encoding="application/x-tex">U_1, U_2, \ldots</annotation></semantics></math>$ of $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>$ that admits no finite subcover. We put $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>V</mi> <mi>n</mi></msub><mo>=</mo><msub><mi>U</mi> <mn>1</mn></msub><mo>∪</mo><mi>…</mi><mo>∪</mo><msub><mi>U</mi> <mi>n</mi></msub></mrow><annotation encoding="application/x-tex">V_n = U_1 \cup \ldots \cup U_n</annotation></semantics></math>$ so that $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>V</mi> <mn>1</mn></msub><mo>⊆</mo><msub><mi>V</mi> <mn>2</mn></msub><mo>⊆</mo><mi>…</mi></mrow><annotation encoding="application/x-tex">V_1 \subseteq V_2 \subseteq \ldots</annotation></semantics></math>$ ; since every point belongs to some $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>V</mi> <mi>n</mi></msub></mrow><annotation encoding="application/x-tex">V_n</annotation></semantics></math>$ but no $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>V</mi> <mi>n</mi></msub></mrow><annotation encoding="application/x-tex">V_n</annotation></semantics></math>$ is all of $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>$ , we may discard repetitions and assume without loss of generality that all the inclusions
∅=V0⊂V1⊂V2⊂…

are strict. Thus for each $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>n</mi><mo>≥</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">n \geq 1</annotation></semantics></math>$ we may pick a point $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>x</mi> <mi>n</mi></msub><mo>∈</mo><msub><mi>V</mi> <mi>n</mi></msub><mo>∖</mo><msub><mi>V</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub></mrow><annotation encoding="application/x-tex">x_n \in V_n \setminus V_{n-1}</annotation></semantics></math>$ . Observe that if $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>m</mi><mo>&lt;</mo><mi>n</mi></mrow><annotation encoding="application/x-tex">m \lt n</annotation></semantics></math>$ , then $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>x</mi> <mi>n</mi></msub><mo>∉</mo><msub><mi>V</mi> <mi>m</mi></msub></mrow><annotation encoding="application/x-tex">x_n \notin V_m</annotation></semantics></math>$ .

Since $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>$ is $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>T</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">T_1</annotation></semantics></math>$ (points are closed), the set $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>W</mi> <mi>m</mi></msub><mo>=</mo><msub><mi>V</mi> <mi>m</mi></msub><mo>∩</mo><mo>¬</mo><mo stretchy="false">{</mo><msub><mi>x</mi> <mn>1</mn></msub><mo>,</mo><mi>…</mi><mo>,</mo><msub><mi>x</mi> <mrow><mi>m</mi><mo>−</mo><mn>1</mn></mrow></msub><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">W_m = V_m \cap \neg \{x_1, \ldots, x_{m-1}\}</annotation></semantics></math>$ is an open neighborhood of $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>x</mi> <mi>m</mi></msub></mrow><annotation encoding="application/x-tex">x_m</annotation></semantics></math>$ that does not contain $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>x</mi> <mi>n</mi></msub></mrow><annotation encoding="application/x-tex">x_n</annotation></semantics></math>$ whenever $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>m</mi><mo>&lt;</mo><mi>n</mi></mrow><annotation encoding="application/x-tex">m \lt n</annotation></semantics></math>$ , and does not contain $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>x</mi> <mi>n</mi></msub></mrow><annotation encoding="application/x-tex">x_n</annotation></semantics></math>$ for $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>n</mi><mo>&lt;</mo><mi>m</mi></mrow><annotation encoding="application/x-tex">n \lt m</annotation></semantics></math>$ . Thus every point $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>x</mi> <mi>n</mi></msub></mrow><annotation encoding="application/x-tex">x_n</annotation></semantics></math>$ is open relative to $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>A</mi><mo>=</mo><mo stretchy="false">{</mo><msub><mi>x</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>x</mi> <mn>2</mn></msub><mo>,</mo><mi>…</mi><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">A = \{x_1, x_2, \ldots\}</annotation></semantics></math>$ , i.e., $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math>$ is a discrete subspace.

Finally, any point $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>x</mi><mo>∉</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">x \notin A</annotation></semantics></math>$ belongs to some $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>V</mi> <mi>n</mi></msub></mrow><annotation encoding="application/x-tex">V_n</annotation></semantics></math>$ , and then $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>V</mi> <mi>n</mi></msub><mo>∩</mo><mo>¬</mo><mo stretchy="false">{</mo><msub><mi>x</mi> <mn>1</mn></msub><mo>,</mo><mi>…</mi><msub><mi>x</mi> <mi>n</mi></msub><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">V_n \cap \neg \{x_1, \ldots x_n\}</annotation></semantics></math>$ is an open neighborhood of $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math>$ that doesn’t intersect $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math>$ . Thus $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math>$ is an infinite closed discrete subspace, meaning that $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>$ is not limit point compact. =–
- In the case of metric spaces, countably compact metric spaces are equivalently compact metric spaces.
- CommentRowNumber5.
- CommentAuthorTodd_Trimble
- CommentTimeJul 27th 2018
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Author: Todd_Trimble
Format: MarkdownItexApparently I got it to display just now. Not sure what happened earlier.

Apparently I got it to display just now. Not sure what happened earlier.
- CommentRowNumber6.
- CommentAuthorRichard Williamson
- CommentTimeJul 27th 2018
- (edited Jul 27th 2018)
- PermaLink
Author: Richard Williamson
Format: MarkdownItexHi Todd, probably you have seen [here](https://nforum.ncatlab.org/discussion/8753/geometry-of-physics-basic-notions-of-higher-topos-theory) that major changes have been made to the rendering of nLab pages, so there may be a few gremlins. However, in this case I cannot reproduce the problem (just tried a couple of trivial edits on this page, and both were visible immediately). Please let me know if you see the same problem again, I will look into it.

Hi Todd, probably you have seen here that major changes have been made to the rendering of nLab pages, so there may be a few gremlins. However, in this case I cannot reproduce the problem (just tried a couple of trivial edits on this page, and both were visible immediately). Please let me know if you see the same problem again, I will look into it.
- CommentRowNumber7.
- CommentAuthorTodd_Trimble
- CommentTimeJul 27th 2018
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Author: Todd_Trimble
Format: MarkdownItexOkay, will do. Yes, I am aware that there may be glitches while you're working on the system; I'm just reporting them as they arise. Thanks!

Okay, will do. Yes, I am aware that there may be glitches while you’re working on the system; I’m just reporting them as they arise. Thanks!
- CommentRowNumber8.
- CommentAuthorRichard Williamson
- CommentTimeJul 27th 2018
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Author: Richard Williamson
Format: MarkdownItexAbsolutely, it is very helpful that you report them. This issue is a bit mysterious to me for the moment, so I will be very grateful if anyone else who experiences it can also report it. Thanks for your understanding!

Absolutely, it is very helpful that you report them. This issue is a bit mysterious to me for the moment, so I will be very grateful if anyone else who experiences it can also report it. Thanks for your understanding!
- CommentRowNumber9.
- CommentAuthorTodd_Trimble
- CommentTimeJul 28th 2018
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Author: Todd_Trimble
Format: MarkdownItexReworded the argument for Proposition 3.2 to make it appear less indirect. <a href="https://ncatlab.org/nlab/revision/diff/countably+compact+topological+space/9">diff</a>, <a href="https://ncatlab.org/nlab/revision/countably+compact+topological+space/9">v9</a>, <a href="https://ncatlab.org/nlab/show/countably+compact+topological+space">current</a>

Reworded the argument for Proposition 3.2 to make it appear less indirect.

diff, v9, current
- CommentRowNumber10.
- CommentAuthorTodd_Trimble
- CommentTimeJul 28th 2018
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Author: Todd_Trimble
Format: MarkdownItexHi Richard. At [[limit point compact space]], the link to [[metrizable space]] is broken. Just reporting. Anything I can do to help?

Hi Richard. At limit point compact space, the link to metrizable space is broken.

Just reporting. Anything I can do to help?
- CommentRowNumber11.
- CommentAuthorRichard Williamson
- CommentTimeJul 28th 2018
- PermaLink
Author: Richard Williamson
Format: MarkdownItexThanks Todd! This is the same issue that [[metrizable space]] has not been rendered by the new renderer. My plan for the moment is remove the restriction on no duplicate redirects, so that we can generate all pages using the new renderer. Then I can switch it back on for edits. Before that, I need to fix an issue with the table of contents rendering that Urs reported. Will keep updated. Sorry for all inconvenience. We are getting there, slowly.

Thanks Todd! This is the same issue that metrizable space has not been rendered by the new renderer. My plan for the moment is remove the restriction on no duplicate redirects, so that we can generate all pages using the new renderer. Then I can switch it back on for edits. Before that, I need to fix an issue with the table of contents rendering that Urs reported. Will keep updated. Sorry for all inconvenience. We are getting there, slowly.

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nLab > Latest Changes: countably compact topological space

Properties

Proposition

Proof

Proposition

Proof