Cardinality of all compact metric spaces

Question

I'm looking for cardinal number of all compact metric spaces. I know that:

1. Cardinal number of compact set is at most $\mathfrak{c}$ (it is a continous image of Cantor set)
2. Compact metric space is separable and complete, so we can look just at countable dense set. It bound our cardinal number to cardinality of $\mathbb{R}^\mathbb{N}$ which is $\mathfrak{c}$

How can I bound it from below?

Answer 1

HINT:

There is no set of all compact metric spaces, because every singleton is a compact metric space, and the collection of all singletons is not a set.

However, note that every compact metric space is homeomorphic to a closed subspace of the Hilbert cube. So if you are only interested in equivalence classes of compact metric spaces, it suffices to consider subspaces of the Hilbert cube.