Let $X$ be a zero-dimensional subset of the plane $\mathbb R ^2$. Is $\mathbb R ^2\setminus X$ necessarily path-connected? I feel the answer must be yes but I need a reference. If it helps, assume $X$ is nowhere dense.



Is the complement of a zero-dimensional subset of the plane path-connected?
If the zero-dimensional set $X$ is not closed, then the answer is "no".


To construct a suitable example, take any open bounded neighborhood $U\subset\mathbb R^2$ of zero, whose boundary $\partial U$ does not contain a topological copy of $[0,1]$. For example, for $U$ we can take a bounded connected component of the complement of the union of two suitable pseudoarcs in the plane. Then the set $$X=\mathbb R^2\setminus\{\vec a+\tfrac1n\partial U:\vec a\in\mathbb Q^2,\;n\in\mathbb N\}$$ will have the desired property: it is zero-dimensional and its complement $\mathbb R^2\setminus X$ does not contain a copy of $[0,1]$ (by the Baire Theorem) and hence is not path-connected.


