Locally compact Hausdorff implies completely regular: Difference between revisions

Revision as of 15:17, 25 October 2009

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 completely regular.

Related facts

Facts used

Any locally compact Hausdorff space possesses a one-point compactificatoin; hence, it is a subset (with the subspace topology) of a compact Hausdorff space.
Compact Hausdorff implies normal
Normal implies completely regular
Complete regularity is hereditary

Proof

Given: A locally compact Hausdorff space $X$ .

To prove: $X$ is a completely regular space.

Proof:

$X$ is a subspace of a compact Hausdorff space $Y$ : This follows from fact (1); we can take $Y$ to be the one-point compactification of $X$ .
$Y$ is a normal space: This follows from fact (2).
$Y$ is a completely regular space: This follows from the previous step and fact (3).
$X$ is a completely regular space: This follows from the previous step and fact (4).

@@ Line 4: / Line 4: @@
 Any [[locally compact Hausdorff space]] is [[completely regular space|completely regular]].
+==Related facts==
+* [[Compact Hausdorff implies normal]]
+* [[Paracompact Hausdorff implies normal]]
+==Facts used==
+# Any locally compact Hausdorff space possesses a [[one-point compactificatoin]]; hence, it is a subset (with the [[subspace topology]]) of a [[compact Hausdorff space]].
+# [[uses::Compact Hausdorff implies normal]]
+# [[uses::Normal implies completely regular]]
+# [[uses::Complete regularity is hereditary]]
 ==Proof==
-The proof is as follows:
+'''Given''': A locally compact Hausdorff space <math>X</math>.
+'''To prove''': <math>X</math> is a completely regular space.
-* Any [[locally compact Hausdorff space]] possesses a [[one-point compactification]]: hence, it is a subset of a compact Hausdorff space
+'''Proof''':
-* Any [[compact Hausdorff implies normal|compact Hausdorff space is normal]], so the space we started with is a subspace of a normal space
-* Any normal space is [[completely regular space|completely regular]], and [[complete regularity is hereditary|any subspace of a completely regular space is completely regular]], so the space we started with is completely regular.
-Another way of viewing this reasoning is to view locally compact Hausdorff spaces as spaces possessing a [[one-point compactification]], and [[completely regular space]]s as spaces that possess ''some'' [[compactification]]; thus, locally compact Hausdorff spaces are completely regular.
+# <math>X</math> is a subspace of a compact Hausdorff space <math>Y</math>: This follows from fact (1); we can take <math>Y</math> to be the one-point compactification of <math>X</math>.
+# <math>Y</math> is a normal space: This follows from fact (2).
+# <math>Y</math> is a completely regular space: This follows from the previous step and fact (3).
+# <math>X</math> is a completely regular space: This follows from the previous step and fact (4).