A complete metric space, or a locally compact Hausdorff space, cannot be written as a countable union of nowhere-dense sets. Named for Rene-Louis Baire, it is a fundamental existence tool in analysis and topology, underlying the open mapping and uniform boundedness theorems.
Facts
StatementEvery completely metrizable topological space is a Baire space, meaning every countable intersection of dense open sets in it is dense. 1 Classification
Statement FormCharacterization Theorem 1 Connections
Has Statement Form
In Branch
Proved By
Source Baire Category Theorem (Wikipedia)
Sources
1. Baire Category Theorem (Wikipedia)
Wikimedia FoundationStatement section, BCT1
Every completely metrizable topological space is a Baire space. More generally, every complete pseudometric space is a Baire space.
lead section, history sentence
Versions of the Baire category theorem were first proved independently in 1897 by Osgood for the real line R and in 1899 by Baire for Euclidean space R^n.
Proved By: Felix Hausdorff, Lead
The more general statement for completely metrizable spaces was first shown by Hausdorff in 1914.
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