Locally compact Hausdorff implies Baire: Difference between revisions
(New page: {{topospace property}} ==Statement== Any locally compact Hausdorff space is a Baire space. ==Proof== The proof follows by combining three facts: * [[Compact Hausdorff implies ...) |
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The proof follows by combining three facts: | The proof follows by combining three facts: | ||
* [[Compact Hausdorff implies Baire| | * [[Compact Hausdorff implies Baire|Any compact Hausdorff space is Baire]] | ||
* A locally compact Hausdorff space can be embedded as an [[open subset]] of a [[compact Hausdorff space]], namely the [[one-point compactification]] | * A locally compact Hausdorff space can be embedded as an [[open subset]] of a [[compact Hausdorff space]], namely the [[one-point compactification]] | ||
* [[Baire is open subspace-closed|Any open subset of a Baire space is a Baire space]] | * [[Baire is open subspace-closed|Any open subset of a Baire space is a Baire space]] | ||
Revision as of 20:59, 23 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
Statement
Any locally compact Hausdorff space is a Baire space.
Proof
The proof follows by combining three facts:
- Any compact Hausdorff space is Baire
- A locally compact Hausdorff space can be embedded as an open subset of a compact Hausdorff space, namely the one-point compactification
- Any open subset of a Baire space is a Baire space