Pseudotopological space

Definition
A pseudotopological space (also called pspace) is a set $$X$$ along with a relation called convergence between $$X$$ and the set of all ultrafilters on $$X$$, such that for any $$x \in X$$, the principal ultrafilter $$\eta_x$$ (i.e., the ultrafilter of all subsets containing $$x$$) is related to the point $$x$$.

If an ultrafilter $$\varphi$$ and a point $$x$$ are related, we say that $$\varphi$$ converges to $$x$$.