The remainder of a Cantor set in the plane Assume  That $M$  is  a  compact  subset of $\mathbb{R}^{2}$ which has zero measure and is homeomorphic to the Cantor  set:
Is the topology or  homology of $\mathbb{R}^{2} \setminus M$  independent of $M$. We have the same question for the relative homology $H_{*}(\mathbb{R^{2}},\mathbb{R}^{2} \setminus M )$?

Non-compact surfaces without boundary are completely classified; see this answer. Then the only relevant data to the homeomorphism type of $\mathbb R^2 \setminus M$ is its space of ends. I claim this is $M \sqcup \{\infty\}$.
This follows because the end compactification of $\Bbb R^2 \setminus M$ is $S^2$. All one needs to do, roughly, is verify that each of the points in $M$ (and $\infty$) is an end of $\Bbb R^2 \setminus M$. This follows then because $M$ is totally disconnected.
