# Baire Space iff Open Sets are Non-Meager

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## Theorem

Let $T = \struct {S, \tau}$ 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.