Timeline for Questions about existence of injections between infinite sets and the sets of all infinite topologies on them
Current License: CC BY-SA 4.0
10 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Mar 25, 2020 at 13:48 | answer | added | Maxime Ramzi | timeline score: 2 | |
| Mar 25, 2020 at 13:38 | comment | added | YCor | @MaximeRamzi thanks! (of course you mean iff it's empty or in the ultrafilter). BTW the number of metrizable topologies is $=2^{|X|}$: the upper bound just consists in counting the number of possible distances. Also if you fix $x_0$ in $X$ and consider, for a nonprincipal ultrafilter $\xi$ on $X-\{x_0\}$, open subsets as finite subsets of $X-\{x_0\}$ and subsets containing $x_0$ and intersecting $X-\{x_0\}$ in an element of $\xi$, one gets $2^{2^{|X|}}$ Hausdorff topologies. | |
| Mar 25, 2020 at 13:32 | comment | added | Maxime Ramzi | @YCor : the number of ultrafilters on $X$ is $2^{2^{|X|}}$, and there is an injection $\beta X\to T_X$ ( a subset is open iff it is in the ultrafilter) | |
| Mar 24, 2020 at 11:43 | history | became hot network question | |||
| Mar 24, 2020 at 9:24 | comment | added | YCor | For every subset $Y$ of $X$ with $|Y|\le |X|$, there exists a metrizable topology on $X$ for which the set of accumulation points is exactly $Y$. Hence $X$ has $\ge 2^{|X|}$ topologies. This is enough to answer the question. [Each topology being an element of $2^{2^X}$, the upper bound is $2^{2^{|X|}}$, so it would be more interesting to ask whether the number of topologies on $X$ is $2^{2^{|X|}}$. For instance I remember once checking that the number of group topologies on the symmetric group $S_\omega$ is $2^{2^{2^{\aleph_0}}}$.] | |
| Mar 24, 2020 at 8:54 | answer | added | Peter LeFanu Lumsdaine | timeline score: 6 | |
| Mar 24, 2020 at 8:29 | answer | added | Wlod AA | timeline score: 4 | |
| Mar 24, 2020 at 4:37 | answer | added | Keith Kearnes | timeline score: 11 | |
| Mar 24, 2020 at 3:43 | history | edited | user153451 | CC BY-SA 4.0 |
added 1 character in body
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| Mar 24, 2020 at 3:36 | history | asked | user153451 | CC BY-SA 4.0 |