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?

It's wiiiiiiide open to compute it exactly. As far as I know the "feature that makes it intractable" is that there's no real feature that makes it tractable. Very broadly speaking, if you want to count the ways that a generic type of structure can be put on an nelement set, there's no efficient way to do this  you basically have to enumerate the structures one by one. This is essentially because "given a description of a structure type and $n$, count the number of structures on an nelement set" is a ridiculously broad problem which ends up reducing to lots and lots of different counting and decision problems. Alternatively I think you can argue via Kolmogorov complexity and all that, but that's not my style? So in any case, the burden of proof is on the person who claims that an efficient counting algorithm should exist. (If you believe some crazy things about complexity theory, like P = PSPACE, this starts to become less true since the structure types hard to enumerate will usually be hard to describe. But if you believe that you're a lost cause in any case :P) It's still reasonable to ask for further justification, though. I'd attempt to give some, but I've been awake for like 30 hours and it would be even more handwavey than the above. The short version: If you do enough enumerative combinatorics, you start to see that nice formulas for enumerating structures arise from one of a few situations. Some really big ones are:
So my answer boils down to: It's intractable 'cause it is. Not a particularly satisfying reason, but sometimes that's the way combinatorics works. Sorry if you read that whole post  I meant for it to be shorter and have more content, but it ended up like most tales told by idiots. But hopefully you learned something, or at least had fun with it? 


I'll expand a little on Harrison's answer. There are several techniques in algebraic combinatorics that allow for exact enumeration of unordered structures; they include
Since finite topologies are an algebraic structure of sorts  a set closed under two operations  you might like to think about a correspondingly difficult algebraic problem, which is the enumeration of finite groups. 


The best result I know is found in the article The number of finite topologies, by D. Kleitman and B. Rothschild, where they state that the base2 logarithm of the number of distinct topologies on a set of $n$ elements is asymptotic to $n^2/4$. 


Sloane's encyclopedia gives some references. I think that if something moves in this question, then Sloane's site will be one of the first to be updated... I am not sure in how far this has to do the UnionClosed Sets Conjecture  the latter is just an extremalcombinatorics style assertion. I don't see how to use it for an enumeration. 


This question is closely related to the UnionClosed Sets Conjecture which is an open question proposed in 1979. 

