Baire Space iff Open Sets are Non-Meager

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Theorem

Let $T = \left({S, \tau}\right)$ be a topological space.


Then $T$ is a Baire space if and only if every non-empty open set of $T$ non-meager in $T$.


Proof


Historical Note

This result was the original definition which René-Louis Baire gave for the Baire space.

The more modern approach is to define it directly in terms of interiors of countable unions of closed sets.