Here’s a fun question: how many different topologies can be defined on a finite set containing n elements? I googled it, and found that the question’s typically posed as “how many different (up to homeomorphism) topologies…”.

It appears that there isn’t a simple answer (that anyone knows of) - it’s a bit like trying to enumerate the n-ary polyominoes, with the “up to homeomorphism” clause having a similar effect to the exclusion from the count of rotations and reflections of the “same” polyomino. I wonder if the two problems are any more deeply related than that.