How many non-homeomorphic topological structures are there on a finite set of order $n$?

Question

Given a finite set of order $n$, how many different (that is, non-homeomorphic) topological structures are there on this set?

It is a question about topology but my feeling is that it is essentially a question in combinatorics.

Answer 1

A topology $\tau$ on the set $X$ is (generated by) a set of subsets of $X$. So $\tau\in P(P(X))$.

Let $\mid X\mid=n$. Then $\mid P(P(X))\mid =2^{{2^n}}$. This gives (at least) an upper bound on the number.

Since the subset must contain the $\emptyset$ and the whole set X, the inequality is strict.

Because of this, we should be able to (marginally, to use @Arturo Magidin's description ) improve it to $2^{2^n-2}=\frac{2^{2^n}}4$.