# How many different topologies can X have?

## Main Question or Discussion Point

Today I was reading some introductory book about topology, and a doubt come to my mind:
If $$X$$ is a finite set of $$n$$ elements, is there a way to know how many different topologies can $$X$$ have?
I think it is some combinatorial problem, but not sure.

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Each topology on X is a subset of P(X), the power set of X. Therefore the set of all topologies on X is a subset of P(P(X)), and a (very, very loose) upper bound on the number of topologies on a set X is $$2^{2^{|X|}}$$. You can find more information here.

This is a question i have also been wondering when i first learned what a topology on a set is. Determining the number of topologies on a finite set, does not seem to be an easy problem, and i am not sure that it is even possible, for the sole fact that not every subset of P(X) (assuming our (X,T) is a topological space) is a topology on X.

I feel like there is a somewhat pseudo-random pattern.

This is so cool! It is interesting how stirling numbers show up in so many places.

However, like they say there, there doesn't seem to be an easy way of counting the number of topologies on a random set of cardinality n. Since T_0 is a well-behaved topology, it seems somewhat easier.

Does anybody know whether this is an Open Question or?

It seems it turned to be a difficult and interesting question.