Baire space

This article defines a property of topological spaces: a property that can be evaluated to true/false for any topological space|View a complete list of properties of topological spaces

Definition

A topological space is termed a Baire space if it satisfies the following equivalent conditions:

• A countable intersection of open dense subsets is dense
• A countable union of closed nowhere dense subsets is nowhere dense

Relation with other properties

Stronger properties

• Compact Hausdorff space
• Locally compact Hausdorff space
• Completely metrizable space

Metaproperties

Hereditariness on open subsets

This property of topological spaces is hereditary on open subsets, or is open subspace-closed. In other words, any open subset of a topological space having this property, also has this property