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
| 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)