Baire space: Difference between revisions
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===Stronger properties=== | ===Stronger properties=== | ||
* [[Compact Hausdorff space]] | * [[Compact Hausdorff space]]: {{proofat|[[compact Hausdorff implies Baire]]}} | ||
* [[Locally compact Hausdorff space]] | * [[Locally compact Hausdorff space]]: {{proofat|[[locally compact Hausdorff implies Baire]]}} | ||
* [[Completely metrizable space]] | * [[Completely metrizable space]]: {{proofat|[[completely metrizable implies Baire]]}} | ||
==Metaproperties== | ==Metaproperties== | ||
{{open subspace-closed}} | {{open subspace-closed}} | ||
Revision as of 02:26, 24 January 2008
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