Number of homeomorphism types of separable closed subspaces of $\beta \mathbb N$.

Solution 1:

There are exactly $2^c$ pairwise nonhomeomorphic separable closed subspaces in $\beta \mathbb N$.

The proof is essentially just a reference to a great paper:

Since $|\beta \mathbb N| = 2^c$, there are $2^c$ countable subsets in $\beta \mathbb N$, hence $\beta \mathbb N$ has only $2^c$ separable subspaces.

The other, non-trivial, direction is mainly the Main Theorem in A. Dow, A.V. Gubbi, A. Szymanski, "Rigid Stone Spaces within ZFC":
There exist $2^c$ pairwise nonhomeomorphic rigid separable Stone spaces,
where Stone space is (in this paper) an extremally disconnected compact Hausdorff space. As it is well-known, each separable extremally disconnected space can be embedded into $\beta \mathbb N$ (see, for instance, Corollary 3.2 here).

Remark: Perhaps there might be much easier constructions, since here we don't need that the spaces are rigid (= the only autohomeomorphism is the identity).