Suppose that $X$ is a connected compact Hausdorff space with the property that for every finite set $F\subseteq X$ the space $X\setminus F$ is connected. Can we conclude that the covering dimension of $X$ is at least 2? Note: I do not assume that $X$ is path-connected.
This question is somewhat dual to the classical problem of Menger whether adjoining finitely many points to a zero-dimensional space keeps the dimension 0.
On
The Bucket-handle continuum is a subset of the plane of dimension $1$, and has no finite separator.
The complement of any finite set is, however, very far from path-connected, in contrast to the Menger sponge.
I wonder if there is a set in the plane of dimension $1$ such that every co-finite subset is path-connected?
The Menger sponge is a counterexample. It is compact, Hausdorff, and of dimension 1, and the complement of any finite subset is path connected.