Here's a set-point topology question I thought of.
How many topological spaces with $n$-points are path-connected?
2026-03-26 16:05:39.1774541139
How many topological spaces with finitely many points are path-connected?
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See this text by May for a proof that for finite topological spaces, path-connectedness and connectedness are equivalent.
I also believe there is no known formula for the number (non-homeomorphic or total) of such topologies on $n$ points as a formula in $n$. A search on the online encyclopedia didn't yield promising results so far.