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: For full proof, refer: compact Hausdorff implies Baire
• Locally compact Hausdorff space: For full proof, refer: locally compact Hausdorff implies Baire
• Completely metrizable space: For full proof, refer: completely metrizable implies Baire

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

For full proof, refer: Baire property is open subspace-closed

References

Textbook references

• Topology (2nd edition) by James R. Munkres^{More info}, Page 296 (formal definition)