What is the cardinality of the set of all ultrafilters containing a Fréchet filter?

Question

Let $\mathfrak{F}$ be the Fréchet filter on an infinite set $X$. $~$What is the cardinality of the set of all ultrafilters containing $\mathfrak{F}\,$? $~$Is it equal to the cardinality $2^{2^{|X|}}$ of the set of all ultrafilters on the set $X\,$?

Answer 1

Yes. If an ultrafilter $U$ does not contain the Fréchet filter, then $U$ must be principal. A principal ultrafilter is determined by its generating element, so there are $|X|$-many principal ultrafilters. Since $|X|<2^{2^{|X|}}$, the cardinality of the set of non-principal ultrafilters must be $2^{2^{|X|}}$.