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Baire category theorem

In mathematics, the Baire category theorem is an important tool in the study of complete spaces, such as Banach spaces and Hilbert spaces, that arise in topology and functional analysis.

The statement is:

Every complete metric space is a Baire space.

The proof of the Baire category theorem uses the axiom of choice; in fact, the Baire category theorem is logically equivalent to a weaker version of the axiom of choice called the axiom of dependent choice.

The Baire category theorem is used in the proof of the open mapping theorem.

Referenced By

Analysis (math) | Analysis (mathematics) | AxiomOfChoice | Axiom of Choice | Baire space | Banach-Schauder theorem | Bounded linear map | Bounded linear operator | Bounded operator | Complete (topology) | Complete metric | Complete metric space | Complete space | Completeness (topology) | Completion (topology) | Continuous operator | First category | Fréchet space | Irrational number | List of functional analysis topics | List of general topology topics | List of mathematical topics | List of mathematical topics (A-C) | List of mathematics topics | Local compactness | Locally compact | Locally compact space | Mathematical analysis | Meager set | Meagre set | Morse theory | Nowhere dense | Open mapping theorem | Operator norm | Pathological (mathematics) | Second category | TopOlogy

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This article is licensed under the GNU Free Documentation License. It uses material from the Wikipedia article "Baire category theorem".

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