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1. • CommentRowNumber1.
   • CommentAuthorTodd_Trimble
   • CommentTimeJul 25th 2018
   • PermaLink
   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

2. • CommentRowNumber2.
   • CommentAuthorTodd_Trimble
   • CommentTimeJul 27th 2018
   • PermaLink
   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

3. • CommentRowNumber3.
   • CommentAuthorTodd_Trimble
   • CommentTimeJul 27th 2018
   • PermaLink
   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.

4. • CommentRowNumber4.
   • CommentAuthorTodd_Trimble
   • CommentTimeJul 27th 2018
   • PermaLink
   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 $X$ 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.
5. • CommentRowNumber5.
   • CommentAuthorTodd_Trimble
   • CommentTimeJul 27th 2018
   • PermaLink
   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.

6. • 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.

7. • CommentRowNumber7.
   • CommentAuthorTodd_Trimble
   • CommentTimeJul 27th 2018
   • PermaLink
   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!

8. • CommentRowNumber8.
   • CommentAuthorRichard Williamson
   • CommentTimeJul 27th 2018
   • PermaLink
   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!

9. • CommentRowNumber9.
   • CommentAuthorTodd_Trimble
   • CommentTimeJul 28th 2018
   • PermaLink
   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

10. • CommentRowNumber10.
    • CommentAuthorTodd_Trimble
    • CommentTimeJul 28th 2018
    • PermaLink
    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?

11. • 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.