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.