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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.

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See and the references therein. – Robert Israel Feb 17 '13 at 20:37
This link maybe helpful for you: – Anirban Oct 24 '14 at 12:40
In case you are interested and didn't know your question is equivalent to: how many preorders exists on a finite set. For any topological space $(X,\tau)$ you can define $x\leq y$ if and only $x \in U \Rightarrow y \in U$ giving a preorder $(X,\leq)$. Conversely given a preorder $(X,\leq)$ one can define a topology on $X$ by setting $U \subseteq X$ open if $x \in U \wedge x \leq y \Rightarrow y \in U$ (i.e. $U$ is up closed). Restricting to finite topological spaces and finite preorders we find that these maps are inverse to each other. – Nex Nov 3 at 10:33

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