Pseudotopological space

This is a variation of topological space. View other variations of topological 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 $x$.