TITLE: Number of valid topologies on a finite set of n elements
QUESTION [41 upvotes]: I've heard that the problem of counting topologies is hard, but I couldn't really find anything about it on the rest of the internet.  Has this problem been solved?  If not, is there some feature that makes it pretty much intractable?

REPLY [3 votes]: You can find the latest results here: https://arxiv.org/pdf/1503.08359.pdf