Q:

Cardinality of the stone Cech compactification of discrete topological spaces

Let $X$ be an infinite set equiped with the discrete topology. If $\beta(X)$ is the stone Cech compactification of $X,$ then is the relation $card(\beta(X))=2^{2^{card(I)}} $ holds?  (this claim is true for $X=\Bbb N.$)

A:

Yes. Indeed for a discrete space $X$, $\beta X$ is just the set of ultrafilters on $X$, and this has cardinality $2^{2^{|X|}}$ if $X$ is infinite.
For a proof of the latter statement, you can see one here, or here.

