I am interested in the relation between global and local homogeneity of topological spaces. On one extreme we have *rigid* spaces, i.e., topological spaces with trivial homeomorphism group.

**Question.** Does there exist a rigid topological space $X$ such that any two points $x,y\in X$ have homeomorphic neighborhoods $O_x,O_y$ (and moreover a homeomorphism $h:O_x\to O_y$ can be chosen so that $h(x)=y$)?