I gained some insight recently when I read an intuitive explanation for what "rich" topologies actually model. It said, basically, that a rich enough topology (say, a Hausdorff one) is a model for "zooming in" indefinitely.
Of course, given the conventions of the subject, I have to assume that a "zoom in" operation would be given by "pointing" at a neighborhood inside the "current" open set.
So what happens if we treat the closed sets as our primary objects? I have in mind the diagram:
$$ \require{AMScd} \begin{CD} \tau @>\cap_X>> \tau\\ @VV'V @VV'V\\ \nu @>\cup_{X'}>> \nu \end{CD} $$
where $\tau$ is a topological space, and $\nu$ is a realization of the equivalent topology defined in terms of closed sets. Does the $\cup_{X'}$ operation correspond, at least intuitively, to zooming out?
Does this intuition extend or generalize for zero-dimensional spaces -- i.e., topologies where every element of $\tau$ is both closed and open?