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What is an example of a topological space $(X,\tau)$ on more than one point, with the following properties?

  1. the only homeomorphism from $X$ to itself is the identity, and
  2. given $x,y\in X$ there are open sets $U, V$ with $x\in U, y\in V$ and a homeomorphism $\varphi: U\to V$ such that $\varphi(x) = y$, where $U,V$ are endowed with their respective subspace topologies.
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    $\begingroup$ Do you want the homeomoprhism from U to V to send x to y in the second condition? As stated, it looks like you can always choose $U=V=X$. $\endgroup$ – Philipp Lampe Feb 13 at 12:28
  • $\begingroup$ Thanks for asking this - I will correct the post! $\endgroup$ – Dominic van der Zypen Feb 13 at 13:12
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    $\begingroup$ Don't you want a Hausdorff space? $\endgroup$ – YCor Feb 13 at 16:03
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    $\begingroup$ @YCor there are Hausdorff (even metrizable) examples. $\endgroup$ – Ramiro de la Vega Feb 13 at 21:32
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For $n\in\mathbb N$ let $U_n=\{m\in\mathbb N:m\geq n\}$. Then $\tau=\{\varnothing\}\cup\{U_n:n\in\mathbb N\}$ is a topology on $\mathbb N$. This space is rigid because $n$ is characterized as the unique element contained in exactly $n+1$ open sets. However, for any $n,m$ the sets $U_n,U_m$ are homeomorphic through a map $k\mapsto k-n+m$ which takes $n$ to $m$.

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There exists a metrizable topological group $H$ such that $H \setminus \{e\}$ is rigid (see Theorem 6.1 in van Mill´s paper: A topological group having no homeomorphisms other than translations).

Exercise: Without knowing anything else about $H$, show that $X=H \setminus\{e\}$ satisfies condition 2 in the OP.

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  • $\begingroup$ Thanks! This is actually a locally connected and connected topological group $H$, which has exponent 2. $\endgroup$ – YCor Feb 13 at 22:06
  • $\begingroup$ @ramirodelavega This exercise kept me busy during a long (and boring) business meeting - thanks! (PS, haven't quite cracked it yet.) $\endgroup$ – Dominic van der Zypen Feb 14 at 7:58

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