Given a collection of open sets, $S$ in some topology, is there a canonical way to find a maximal subset $S' \subset S$ such that all elements of $S'$ are mutually disjoint? I am particularly looking at the finite case of $|S| < \aleph_0$, but am also interested in generalizing to the infinite case.
If we remove the 'open set' criterion, this problem reduces down to a simpler case of finding a maximal independent subset of a given graph, which is known to be NP hard.
Perhaps the addition of a topological element to this problem makes it more approachable, though I have not been able to make any significant progress towards solving it yet myself. Let me know if you know any lines of attack for this problem.
For a topology on a finite set (or more generally, a topology of finite size) you can just take the minimal non-empty open sets, plus the empty set. These form a maximal (indeed, maximum) family of mutually disjoint open sets.
Another canonical maximal such family is the trivial subtopology, consisting of the empty set and the whole set. This is a silly example, but in the infinite case I don't think that you can get more canonical than that.