Exercise 1779Topology › Metric and normed spacesM1Added 3 Nov 2026

Baire category theorem

#Baire#category#complete

Statement

State the Baire category theorem for complete metric spaces.

Hint

In a complete metric space, a countable intersection of dense open sets is dense.

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Theorem (Baire). Let $(X,d)$ be complete. If $(U_n)$ is a sequence of dense open sets, then $\bigcap_nU_n$ is dense.

Consequence. A complete metric space is not a countable union of nowhere dense sets.

Conclusion. Baire's theorem is a powerful tool in functional analysis (open mapping, closed graph, uniform boundedness).