Timeline for Connected $T_2$-space with $\text{Cont}(X,X)$ not dense in $X^X$
Current License: CC BY-SA 3.0
5 events
| when toggle format | what | by | license | comment | |
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| Nov 23, 2017 at 16:45 | history | edited | Luc Guyot | CC BY-SA 3.0 |
Fixes typo: $z$ belongs to the interior of $Z$
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| Nov 23, 2017 at 14:31 | comment | added | YCor | Say that $X$ is approximately path-connected if the graph of the equivalence "be in the same path component" is dense in $X^2$. The argument extends to show that if $X$ is not approximately path-connected, and has at least a nontrivial arc, then the continuous self-maps of $X$ are not dense in $X^X$. For instance the closure of the graph $x\mapsto\sin(1/x)$, $x\ge 0$, is approximately path connected, and so are solenoids, since they have a dense path-connected subset. | |
| Nov 23, 2017 at 14:26 | vote | accept | Dominic van der Zypen | ||
| Nov 23, 2017 at 14:22 | history | edited | YCor | CC BY-SA 3.0 |
added 18 characters in body
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| Nov 23, 2017 at 14:17 | history | answered | YCor | CC BY-SA 3.0 |