Baire category theorem

The Baire category theorem (BCT) is an important tool in general topology and functional analysis. The theorem has two forms, each of which gives sufficient conditions for a topological space to be a Baire space.

The theorem was proved by René-Louis Baire in his 1899 doctoral thesis.

A Baire space is a topological space with the following property: for each countable collection of open dense sets ${\displaystyle \{U_{n}\}_{n=1}^{\infty }}$, their intersection ${\displaystyle \textstyle \bigcap _{n=1}^{\infty }U_{n}}$ is dense.

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