The Number of Topologies on a Finite Set

I would like to know if there is like a magical formula to know how many topologies exist on a finite set

For example for $X = \{ a, b, c \}$ I found $29$, but I dont know if there are more or how to know this exact number without writing all topologies first.

Solution 1:

The number 29 is apparently correct. See oeis. It is the same as the number of quasi orders on the set, as well as some other equivalent formulations. I don't know any quick way to compute it; maybe because I can tell a doughnut from a coffee cup. ..

All kidding aside (or almost all ), consider a set with $5$ labeled elements. Then there are $2^5$, or $32$ subsets to "work with "...

Now, we need to see how many ways some of these can be selected which meet the definition of a topology...

Namely, $\emptyset $ and the whole set, call it $X $; and closure under finite (which is automatic, here) intersections, and arbitrary unions...

There are, amazing in a way, $6942$ ways to do this...

If you think of how many ways you can choose subsets, you're looking at the power set again, so $2^{2^5} $ ways... Now, $2^{10}\approx 10^3$, so we're looking at on the order of $10^9$ collections that we can use to try to build a topology. ..

Only about $1$ in every $10^6$ subsets of the power set qualifies as a topology. ..

The moral of the story seems to be that we have our work cut out for us... and remember we are only at $n=5$...