David R. MacIver's Blog

More finite topological spaces

In fact, for any finite topological space \(X\) we have that \(w(X) = |X|\).

Proof: For any point \(x\) define \(U_x\) to be the intersection of all open sets containing \(x\). The set \({ U_x : x in X }\) is a basis for the topology of \(X\) of cardinality \(|X|\).

See here for far more details.