Suppose you have two pseudo-topological spaces $(X,p_1)$ and $(X,p_2)$ where $p_1$ and $p_2$ are relations between the set of ultrafilters on $X$ and the points in $X$.
We can define a topological space using the relation $p_1$ or $p_2$ by saying that $U\subseteq X$ is open if $\forall x\in U $ whenever an ultrafilter $F\rightarrow x$ (i.e. the relation contains $(F,x)$ pair) we have $U\in F$.
If $p_1 \neq p_2$ then are the generated topologies different? Clearly if the pseudo-topological space is topological - that is ultrafilters converge in the pseudotopological space iff they converge in the topological space - then the generated topologies are unique.
But not every pseudo-topological space is topological and I can't seem to show if pseudo-topologies always generate unique topologies. Is it true?