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Let $\kappa$ in an uncountable regular cardinal and $X$ be a space and $e(x)=\kappa$, where the ``extent'' $e(X)$ of $X$ is the supremum of the cardinalities of closed discrete subsets of $X$. My question is this:

Under what condition, the space $X$ contains a closed discrete subset $Y$ such that $|Y|=\kappa$? It is known that if $X$ is metrizable, then we may find such subspace of $X$. What about if $X$ is a Moore space or other generalized metrizable spaces.

Thanks a lot.

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    $\begingroup$ It's not true if $\kappa=\omega$ and $X$ is, say, Hausdorff compact infinite, so even in the metrizable case it's not always true. Maybe you have in mind $\kappa$ uncountable. $\endgroup$
    – YCor
    Commented May 11, 2019 at 9:39
  • $\begingroup$ Remark: If $\kappa$ any infinite cardinal, take a set $Y$ of cardinal $\kappa$, $X=Y\sqcup\{x_0\}$ with the Hausdorff topology for which closed subsets subsets containing $x_0$ and all subsets whose intersection with $Y$ has cardinal $<\kappa$. It has extent $\kappa$ and is Hausdorff, and the extent is not attained. $\endgroup$
    – YCor
    Commented May 11, 2019 at 9:43
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    $\begingroup$ PS in my last example I should assume that $\kappa=\aleph_\lambda$ with $\lambda=0$ or $\lambda$ limit cardinal (to ensure that the extent is $\kappa$). This is a necessary condition since obviously if the extent is $\aleph_{\alpha+1}$ then it's attained. $\endgroup$
    – YCor
    Commented May 11, 2019 at 10:13
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    $\begingroup$ @YCor, your example is metrizable if $cf(\kappa) =\omega$, so we actually need to assume $cf(\kappa) > \omega$ in the result about metric spaces. $\endgroup$
    – Santi Spadaro
    Commented May 11, 2019 at 11:00
  • $\begingroup$ @SantiSpadaro yes you're right indeed, this is a counterexample to the OP's claim in this case. A metric on $X$ comes as follows: write $X=\bigsqcup_nX_n$, with $|X_n|<\kappa$, and write $n=\nu(x)$ if $x\in X_n$ and $\nu(x_0)=\infty$. Take the ultrametric $d(x,y)=2^{-\min(\nu(x),\nu(y)}$: it defines the topology. $\endgroup$
    – YCor
    Commented May 11, 2019 at 11:09

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