On pg 76 of Jech's Set Theory, he proves the existence of a nonprincipal ultrafilter on $\omega$ that is not a $p$-point.

Given a partition $\{A_n\}$ of $\omega$ into $\aleph_0$ infinite pieces, let $F$ be the following filter on $\omega$.

$X \in F$ if and only if except for finitely many $n$, $X \cap A_n$ contains all but finitely many elements of $A_n$.

If $D$ is any ultrafilter extending $F$, then $D$ is not a $p$-point.

I've been thinking why the ultrafilter $D$ isn't a $p$-point but I couldn't figure it out

a compact Hausdorff space with only P-points is finite[Proof:otherwise, the set of open subsets is closed under countable intersections, and hence all countable subsets are closed. In turn this implies that all countable subsets are discrete (since removing an accumulation point would yield a non-closed countable subset), and since a discrete compact set is finite, this implies that all countable subsets are finite, which obviously implies finite.] $\endgroup$