Let X be a Hausdorff space and Difine the Property A as following: if $\mathscr{U}$ is a collection of open sets of X that witnesses Hausdorff property of X (= $\forall x,y \in X$, there exist two disjoint opensets $U_1$ and $U_2$ $\in \mathscr{U}$, st $x \in U_1$ and $y \in U_2$ ), then there is a point $x\in X$ such that $U \in \mathscr{U}: x \in U>\omega$. Is there a countable pseudocharacter Hausdorff space X with the property A?

Hodel defined the Hausdorff pseudocharacter $H\psi(X)$ as the smallest infinite cardinal so that each point lies in at most $H\psi(X)$ elements of an open family $\mathcal{U'}$ that witnesses the Hausdorff property of $X$. Any $X$ such that $\omega = \psi(X) < H\psi(X)$ would seem to answer your question. I don't know of any specific example, but I guess there must be one. Why define $H\psi$ if it's really the same as $\psi$? 

