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removed condition, mentioned by Philipp in a comment - thanks!

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 $\varphi(x) = y$.