Timeline for Compactness of $C(X,X)$
Current License: CC BY-SA 4.0
7 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Mar 10, 2020 at 12:23 | vote | accept | Sh.M1972 | ||
| Mar 10, 2020 at 12:08 | comment | added | Alexander Schmeding | @YCor: Thanks for the correction, I am always thinking nice metrisable spaces, i.e. manifolds... | |
| Mar 10, 2020 at 12:07 | history | edited | Alexander Schmeding | CC BY-SA 4.0 |
added 280 characters in body
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| Mar 10, 2020 at 9:54 | comment | added | Sh.M1972 | @YCor so $C(A^G, A^G)$ is not compact, because $A^G$ is totally disconnected ($G$ is f.g). | |
| Mar 10, 2020 at 8:28 | comment | added | YCor | Still for $X$ compact metrizable $C(X,X)$ is non-compact in many cases, notably (a) when $X$ has infinitely many connected components (b) when $X$ contains a non-trivial arc. Also, for $X$ compact metrizable $C(X,[0,1])$ is compact iff $X$ is finite. | |
| Mar 10, 2020 at 7:37 | comment | added | YCor | This seems to be in contradiction with this answer, which says that there exists an infinite connected compact metrizable space, in which every continuous self-map is identity or constant. | |
| Mar 10, 2020 at 7:06 | history | answered | Alexander Schmeding | CC BY-SA 4.0 |