Topological models of modal logic

The previous post discussed a superficial connection between modal logic and topology, that both give use the terms regular and normal to indicate added sets of axioms. McKinsey and Tarski developed a deeper connection between modal logic and topology that we’ll discuss here. Starting with a topological space X and a proposition p, define [[p]] as the set of points in X at which p is true. Define □p to be true at points in the interior of [[p]] and define ◇p to be true on the closure of [[p]]. You could think of □p as the points where p is robustly true. Not only is p true at x, there’s some wiggle room around x, i.e. an open set, in which p remains true. You could think of ◇p as there points where we cannot rule out the possibility of p being true using open sets. If ◇p includes x, any open set containing x also contains part of ◇p, though it may also contain points outside of ◇p. Regularity For any topology on X, the logic constructed above is regular. The axiom ◇p ⇔ ¬ (□ ¬ p) holds…