# 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.

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