I need a space with countable tightness which is not a Fréchet space. In this space, I am searching for a point with **no** deleted neighborhood consisting entirely of P-points.


(A P-point is a point $x \in X$ such that for every $G\_\delta$ set $O$ containing $x$, $x \in \operatorname{int}(O)$ or equivalently $M\_x = O\_x$, i.e every fixed prime z-filter that contains $x$ is a z-ultrafilter.)



A space with countable tightness which is not a Fréchet space?
First note that in a space with countable tightness: $P$-point $\Leftrightarrow$ weak $P$-point $\Leftrightarrow$ Isolated point. So you are looking for a point for which every deleted neighborhood contains a non-isolated point. You can do a lot better:


Let $X$ be a countable *maximal space* (i.e. such that the topology on $X$ is maximal among those topologies which have no isolated points). This is an easy application of Zorn´s lemma, starting with any countable space which has no isolated points. Then:


1) $X$ is countable (in particular it has countable tightness),


2) No point of $X$ is a limit point of two disjoint subsets of $X$ (in particular it has no non-trivial convergent sequences and therefore it is not Fréchet),


3) $X$ has no isolated points (so **any** point has the property that you want).


Property 2 is not so easy to check, but you can look at **"Applications of maximal topologies"** by E.K. van Douwen, where he proves (see Theorem 2.2) that for *crowded* spaces (i.e. spaces with no isolated points) property 2 (which he calls *perfectly disconnected*) is equivalent to being a maximal space. In the same paper, van Douwen shows how you can construct such a space which is also regular (this is not a simple application of Zorn´s lemma anymore).


