Interiors and Closures
When we pick a topology on a set , not every subset is open, and not every subset is closed. However, we can still come up with some open and closed sets from any subset .
For the open set, notice that we always have at least one open set inside : the empty set. So we can gather up all the open sets contained in and take their union. Since they’re all contained in the union will be as well. And since arbitrary unions of open sets are still open, it’s an open set. In fact, it’s the largest open set contained in , because it contains all the other open subsets of . We call this the “interior” of , denoted or . Clearly the interior of an open set it the set itself.
Dually, we know that there is at least one closed set containing : the whole space . Then the intersection of all the closed sets containing will be a closed set containing , and will be the smallest such closed set. We call this the “closure” of and write or . As for the interior, the closure of a closed set is the set itself.
Now the complement of the closure of is an open set contained in the complement of . In fact, any other open set contained in the complement of will be contained in this one, so it is the interior of the complement of . Dually, the closure of the complement of is the complement of the interior of .
We can write this fact down categorically as well. Since it reverses subset containment, complementation is a contravariant equivalence from the poset of subsets of (considered as a category) to itself. That is, is equivalent to . The interior and closure operators are covariant functors from to itself, since they preserve containment. The previous paragraph states that these two functors are dual to each other, in the sense that is the same functor as under the above equivalence. So all the really important information is contained in the closure functor.
Now, what do we know about this functor? Well, since is contained in we have a natural transformation . Then since is contained in we have a natural transformation . I haven’t really covered these yet, but it’s straightforward from here to verify that along with these two natural transformations forms a monad. If you’re interested in learning more right away, go check out The Catsters’ series of YouTube videos.
We also can easily check that , and that . That is, the functor preserves all finite coproducts. It turns out that this is enough to characterize the topology in its entirety!
Given a set , a closure operator on is a monad , where is a functor which preserves finite coproducts. This data is equivalent to the four axioms given by Kuratowski:
From here we can define the closed sets of to be those in the image of the functor . From axiom 1 we see that , but this closure must be a subset of , and so is closed. Axiom 4 tells us straight off that is closed. Axiom 3 tells us that the finite union of closed sets is closed. We just need to know that arbitrary intersections of closed sets are again closed.
For this, we note that given any collection of subsets, the intersection lies in each , and so by functoriality for each . Thus we see that . In particular, if all the are closed, then . But since for all , this inclusion is actually an equality, and thus the intersection of the is in the image of the closure functor. And thus we really have constructed the closed sets of a topology on .