Timeline for Closed and discrete sets
Current License: CC BY-SA 4.0
10 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| May 11, 2019 at 20:12 | comment | added | Henno Brandsma | Look in Juhasz' two books on cardinal functions in topology; he calls this the sup=max problem. | |
| May 11, 2019 at 11:09 | comment | added | YCor | @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. | |
| May 11, 2019 at 11:00 | comment | added | Santi Spadaro | @YCor, your example is metrizable if $cf(\kappa) =\omega$, so we actually need to assume $cf(\kappa) > \omega$ in the result about metric spaces. | |
| May 11, 2019 at 10:13 | comment | added | YCor | 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. | |
| May 11, 2019 at 10:11 | history | edited | user1 | CC BY-SA 4.0 |
added 11 characters in body
|
| S May 11, 2019 at 10:10 | history | suggested | user64494 | CC BY-SA 4.0 |
A typo in the title is corrected.
|
| May 11, 2019 at 9:45 | review | Suggested edits | |||
| S May 11, 2019 at 10:10 | |||||
| May 11, 2019 at 9:43 | comment | added | YCor | 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. | |
| May 11, 2019 at 9:39 | comment | added | YCor | 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. | |
| May 11, 2019 at 8:57 | history | asked | user1 | CC BY-SA 4.0 |