Is it possible that $\Bbb R^2-C$ can be homeomorphic to $\Bbb R^2-U$ where $C$ is countably infinite and $U$ is uncountable?
Intuitively I believe the answer is no, but I'm having difficulty showing this in full generality.
I know that $\Bbb R^2-C$ is always path-connected when $C$ is countable. And I know that $\Bbb R^2-U$ consists of two path-connected components when $U$ is the graph of a continuous function from $\Bbb R$ to $\Bbb R$ or when $U$ is a simple closed curve. But these are very limited cases. I know that if $U$ is uncountable, then uncountably many points of $U$ are limit points of $U$, but that hasn't been helping much.
Any help in answering this general problem is appreciated.