# Rigid space, but with homeomorphic neighborhoods

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.
• 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 $U=V=X$. – Philipp Lampe Feb 13 at 12:28
• Thanks for asking this - I will correct the post! – Dominic van der Zypen Feb 13 at 13:12
• Don't you want a Hausdorff space? – YCor Feb 13 at 16:03
• @YCor there are Hausdorff (even metrizable) examples. – Ramiro de la Vega Feb 13 at 21:32

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 $$m$$.
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.
• Thanks! This is actually a locally connected and connected topological group $H$, which has exponent 2. – YCor Feb 13 at 22:06