Question

Let $X$ be a compact Hausdorff space. A P-point in $X$ is a point which does not lie in the boundary of the cozero set of a continuous real-valued function on $X$.

Question. Suppose that $X$ has no isolated points. Is the set of P-points a meagre subset of $X$? Failing this, is the set of non-P-points at least a Baire space?

Answer 1

The set of non-P-points is a Baire space:

I´ll work with the usual definition of P-point: $p \in X$ is a P-point if the intersection of countably many neighborhoods of $p$ is again a neighborhood of $p$. This is equivalent to the definition in the question for compact $X$.

Fix a compact $X$, let $P \subseteq X$ be the set of all P-points and $N=X \setminus P$. The first thing to notice is that if $A$ is a $G_\delta$ subset of $X$ and $A \subseteq P$ then every $p \in A$ is isolated. This is because (using that $p$ is a P-point and regularity of $X$) there would be a closed (hence compact) neighborhood of $p$ consisting only of P-points. Since compact P-spaces are finite, we get that $p$ is isolated. It follows that if $X$ has no isolated points then $P$ contains no $G_\delta$ and in particular $N$ is dense in $X$.

Now if $\left< U_n : n\in \omega \right>$ is a sequence of open dense subsets of $N$, then each $U_n=V_n \cap N$ for some $V_n$ open in $X$. Since $N$ is dense in $X$ we have that $V_n$ must also be dense in $X$. Then $\bigcap_{n \in \omega}V_n$ is dense in $X$ and since $P$ contains no $G_\delta$, we get that $\bigcap_{n \in \omega}U_n=N \cap \bigcap_{n \in \omega}V_n$ is dense in $N$.