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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{{topospace property}} | {{topospace property implication}} | ||
==Statement== | ==Statement== | ||
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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]] | ||
Latest revision as of 19:48, 11 May 2008
This article gives the statement and possibly, proof, of an implication relation between two topological space properties. That is, it states that every topological space satisfying the first topological space property must also satisfy the second topological space property
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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