Baire space

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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:

Relation with other properties

Stronger properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
compact Hausdorff space compact and Hausdorff compact Hausdorff implies Baire |FULL LIST, MORE INFO
locally compact Hausdorff space locally compact and Hausdorff locally compact Hausdorff implies Baire |FULL LIST, MORE INFO
completely metrizable space arises from a complete metric space completely metrizable implies Baire |FULL LIST, MORE INFO

Weaker properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
Volterra 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

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

References

Textbook references

  • Topology (2nd edition) by James R. MunkresMore info, Page 296 (formal definition)