and $U$ and $V$ are disjoint open subsets of $X$, then is there a homeomorphism $h:X\to X$ such that $h[U]\subseteq V$?
What about such that $h[U]\subseteq V$ and $h[V]\subseteq U$?
2026-04-02 10:49:36.1775126976
If $X$ is a topological group
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This is false even for the additive group of reals, for $V=(-1,1)$ and $U=\Bbb R\setminus [-1,1]$, because $\overline{V}$ is compact whereas $\overline{U}$ is not.