Local vs global homogeneity of topological spaces

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 $h(x)=y$)?

• $X = \mathrm{pt}$ works but is not in the spirit of what you're looking for. – Arun Debray Nov 9 '17 at 4:21
• A much easier particular case: is there such an example for Alexandroff spaces? That is, is there a rigid poset $X$ such that the posets ${\uparrow}x$ are isomorphic for every $x\in X$? – მამუკა ჯიბლაძე Nov 9 '17 at 5:07
• @მამუკაჯიბლაძე I do not know about this particular case neither, but admit that there can exist a finite space with those properties. Then it will be a matter of combinatorics. – Taras Banakh Nov 9 '17 at 8:12
• Well I doubt about finite because if the smallest neighborhood of $x$ contains any $y$ with $x$ not in the smallest neighborhood of $y$, then the smallest neighborhood of $y$ will be forced to have smaller cardinality than that of $x$. So we seemingly end up with a topological sum of antidiscretes, which will not be rigid except in trivial cases. There might however exist some infinite examples. – მამუკა ჯიბლაძე Nov 9 '17 at 8:17
• @მამუკაჯიბლაძე Great! Thank you for the comment due to which the finite case is ruled out. – Taras Banakh Nov 9 '17 at 8:29