Is the nbhood filter of a point $x$ an ultrafilter?
It seems to me that this is true in metric space: If $F_x$ is a nbhood filter no sets can be added to it. If a set $Y$ has non-empty intersection with all nbhoods of $x$ then $x$ is in the closure of the set $Y$. If $Y$ is added the resulting collection is not a filter because all intersections are added and there exists an intersection which is not a nbhood. For example in $\mathbb R$ $x=0$ and $Y=[0,1]$.
In which topological spaces is the nbhood filter an ultrafilter?