# Baire Space iff Open Sets are Non-Meager

From ProofWiki

## 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.