Baire Space is Non-Meager

Theorem

Let $T = \struct {S, \tau}$ be a Baire space (in the context of topology).


Then $T$ is non-meager in $T$.


Proof

From Baire Space iff Open Sets are Non-Meager, all open sets of $T$ are non-meager in $T$.

But $T$ itself is an open set of $T$ by definition of topological space.

Hence the result.

$\blacksquare$


Sources

  • 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): Baire space
  • 2020: James C. Robinson: Introduction to Functional Analysis ... (previous) ... (next) $22.1$: The Baire Category Theorem
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