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    • CommentRowNumber1.
    • CommentAuthorTodd_Trimble
    • CommentTimeJul 25th 2018

    I added the example of the long line.

    diff, v4, current

    • CommentRowNumber2.
    • CommentAuthorTodd_Trimble
    • CommentTimeJul 27th 2018

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

    diff, v5, current

    • CommentRowNumber3.
    • CommentAuthorTodd_Trimble
    • CommentTimeJul 27th 2018

    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

    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 XX is countably compact and AA is a countable closed discrete subspace. For each aAa \in A, choose an open neighborhood U aU_a such that U aA={a}U_a \cap A = \{a\}, and let V aV_a be the open subset U a¬AU_a \cup \neg A. Clearly we still have V aA={a}V_a \cap A = \{a\}. Also, {V a:aA}\{V_a: a \in A\} is a countable cover of XX, hence admits a finite subcover V a 1,,V a nV_{a_1}, \ldots, V_{a_n}. But then

    A=( i=1 nV a i)A= i=1 n(V a iA)={a 1,,a n}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 1T_1 and limit point compact is countably compact. =–

    +– {: .proof}

    Proof

    Equivalently (under classical logic), the assertion is that if a T 1T_1 space XX 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,U_1, U_2, \ldots of XX that admits no finite subcover. We put V n=U 1U nV_n = U_1 \cup \ldots \cup U_n so that V 1V 2V_1 \subseteq V_2 \subseteq \ldots; since every point belongs to some V nV_n but no V nV_n is all of XX, we may discard repetitions and assume without loss of generality that all the inclusions

    =V 0V 1V 2\emptyset = V_0 \subset V_1 \subset V_2 \subset \ldots

    are strict. Thus for each n1n \geq 1 we may pick a point x nV nV n1x_n \in V_n \setminus V_{n-1}. Observe that if m<nm \lt n, then x nV mx_n \notin V_m.

    Since XX is T 1T_1 (points are closed), the set W m=V m¬{x 1,,x m1}W_m = V_m \cap \neg \{x_1, \ldots, x_{m-1}\} is an open neighborhood of x mx_m that does not contain x nx_n whenever m<nm \lt n, and does not contain x nx_n for n<mn \lt m. Thus every point x nx_n is open relative to A={x 1,x 2,}A = \{x_1, x_2, \ldots\}, i.e., AA is a discrete subspace.

    Finally, any point xAx \notin A belongs to some V nV_n, and then V n¬{x 1,x n}V_n \cap \neg \{x_1, \ldots x_n\} is an open neighborhood of xx that doesn’t intersect AA. Thus AA is an infinite closed discrete subspace, meaning that XX is not limit point compact. =–

    • CommentRowNumber5.
    • CommentAuthorTodd_Trimble
    • CommentTimeJul 27th 2018

    Apparently I got it to display just now. Not sure what happened earlier.

    • CommentRowNumber6.
    • CommentAuthorRichard Williamson
    • CommentTimeJul 27th 2018
    • (edited Jul 27th 2018)

    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

    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!

  1. 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

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

    diff, v9, current

    • CommentRowNumber10.
    • CommentAuthorTodd_Trimble
    • CommentTimeJul 28th 2018

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

    Just reporting. Anything I can do to help?

  2. 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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