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?



non-P-points a Baire space
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 $N$.


