Baire space: Difference between revisions
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===Stronger properties=== | ===Stronger properties=== | ||
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! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions | |||
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| [[Weaker than::compact Hausdorff space]] || [[compact space|compact]] and [[Hausdorff space|Hausdorff]] || [[compact Hausdorff implies Baire]] || || {{intermediate notions short|compact Hausdorff space|Baire space}} | |||
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| [[Weaker than::locally compact Hausdorff space]] || [[locally compact space|locally compact]] and [[Hausdorff space|Hausdorff]] || [[locally compact Hausdorff implies Baire]] || || {{intermediate notions short|locally compact Hausdorff space|Baire space}} | |||
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| [[Weaker than::completely metrizable space]] || arises from a [[complete metric space]] || [[completely metrizable implies Baire]] || || {{intermediate notions short|completely metrizable space|Baire space}} | |||
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===Weaker properties=== | |||
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! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions | |||
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| [[Stronger than::Volterra space]] || || || || | |||
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==Metaproperties== | ==Metaproperties== | ||
{{open subspace-closed}} | {{open subspace-closed}} | ||
{{proofat|[[Baire property is open subspace-closed]]}} | |||
==References== | ==References== | ||
===Textbook references=== | ===Textbook references=== | ||
* {{booklink|Munkres}}, Page 296 (formal definition) | * {{booklink|Munkres}}, Page 296 (formal definition) | ||
Latest revision as of 16:16, 20 October 2010
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)