My question is more or less related to basic set theory. But I don't know even that. Apologies if I added the wrong tags.

How many non-compact surfaces are there upto homeomorphism?

My idea is to produce an uncountable family $\mathcal F$ of closed subsets of the Cantor sets such that any two distinct elements of $\mathcal F$ are non-homeomorphic. Once I show this, the rest follows from the fact below:

Let $\mathcal P_1,\mathcal P_2$ be two non-empty, closed subsets of the Cantor set. Then $\Bbb S^2\setminus \mathcal P_1$ is homeomorphic to $\Bbb S^2\setminus \mathcal P_2$ if and only if $\mathcal P_1$ is homeomorphic to $\mathcal P_2$. This is a very particular case of Kerékjártó's classification theorem of non-compact surfaces.