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Completely metrizable implies Baire

This article gives the statement and possibly, proof, of an implication relation between two topological space properties. That is, it states that every topological space satisfying the first topological space property (i.e., completely metrizable space) must also satisfy the second topological space property (i.e., Baire space)
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Contents

Statement

If a topological space is completely metrizable (i.e. it can be given the structure of a complete metric space) then it is a Baire space.

Definitions used

Completely metrizable space

Further information: Completely metrizable space

Baire space

Further information: Baire space

Proof