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.

Motivation: How many non-compact (planar) surfaces are there upto homeomorphism?

Question: How many pairwise non-homeomorphic non-empty closed subsets of the Cantor set are there?

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.

More generally, I have the following question:

Let $\mathcal F'$ be the collection of all pairs $(\mathcal P,\mathcal A)$, where $\mathcal P$ is a non-empty closed subset of the Cantor set, and $\mathcal A$ is a closed subset of $\mathcal P$. Define an equivalence relation $\sim$ on $\mathcal F'$ as follows: $(\mathcal P_1, \mathcal A_1)\sim (\mathcal P_2, \mathcal A_2)$ if and only if there is a homeomorphism $\varphi\colon \mathcal P_1\to \mathcal P_2$ with $\varphi(\mathcal A_1)=\mathcal A_2$.

Question: What's the cardinality of $\mathcal F'/\sim$?