Locally compact Hausdorff implies completely regular

Statement
Any locally compact Hausdorff space is completely regular.

Related facts

 * Compact Hausdorff implies normal
 * Paracompact Hausdorff implies normal

Facts used

 * 1) Any locally compact Hausdorff space possesses a one-point compactification; hence, it is a subset (with the subspace topology) of a compact Hausdorff space.
 * 2) uses::Compact Hausdorff implies normal
 * 3) uses::Normal implies completely regular
 * 4) uses::Complete regularity is hereditary

Proof
Given: A locally compact Hausdorff space $$X$$.

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

Proof:


 * 1) $$X$$ 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 $$X$$.
 * 2) $$Y$$ is a normal space: This follows from fact (2).
 * 3) $$Y$$ is a completely regular space: This follows from the previous step and fact (3).
 * 4) $$X$$ is a completely regular space: This follows from the previous step and fact (4).