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Oct 3, 2019 at 12:49 vote accept Dominic van der Zypen
Oct 3, 2019 at 9:38 answer added Taras Banakh timeline score: 2
Oct 3, 2019 at 8:24 comment added Dominic van der Zypen I think already the sets $X\setminus \{x\}$ for $x\in X$ pose a challenge, but your interesting idea might work, thanks @TarasBanakh!
Oct 3, 2019 at 8:10 comment added Taras Banakh I hope that one can construct such a rigid space $X$ even as a subspace of the real line. Just modify the construction of a Bernstein set killing all possible homeomorphisms between Borel subsets of $X$ containing no isolated points.
Oct 3, 2019 at 8:06 history edited Dominic van der Zypen CC BY-SA 4.0
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Oct 3, 2019 at 8:01 history asked Dominic van der Zypen CC BY-SA 4.0