Baire category theorem

The Baire category theorem (BCT) is an important result 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 (a topological space such that the intersection of countably many dense open sets is still dense). It is used in the proof of results in many areas of analysis and geometry, including some of the fundamental theorems of functional analysis.

Versions of the Baire category theorem were first proved independently in 1897 by Osgood for the real line ${\displaystyle \mathbb {R} }$ and in 1899 by Baire^[1] for Euclidean space ${\displaystyle \mathbb {R} ^{n}}$.^[2] The more general statement for completely metrizable spaces was first shown by Hausdorff^[3] in 1914.