Locally compact Hausdorff implies completely regular
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.
Proof
The proof is as follows:
- Any locally compact Hausdorff space possesses a one-point compactification: hence, it is a subset of a compact Hausdorff space
- Any compact Hausdorff space is normal, so the space we started with is a subspace of a normal space
- Any normal space is completely regular, and 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 spaces as spaces that possess some compactification; thus, locally compact Hausdorff spaces are completely regular.