More finite topological spaces

In fact, for any finite topological space [tex]X[/tex] we have that [tex]w(X) = |X|[/tex].

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

See here for far more details.

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