This is an interesting question I myself have asked. It turns out that there is no known formula for the number of topologies on a set with finite cardinality, and my guess (though this could be wrong) is that there is not a given formula for infinite sets. In fact, a lot of interesting things about the number of topologies on a finite set (like the number of topologies is the same as the number of preorders, and the number of Kolomogorov topologies is the number of partial orderings) can't be said in general for infinite sets. I'll give this more though and get back to you.