local compactness is hereditary for locally closed subspaces

Theorem - Let $X$ be a locally compact space and $Y\subseteq X$ a subspace. If $Y$ is locally closed in $X$ then $Y$ is also locally compact.

The converse of this theorem is also true with the additional assumption that $X$ is Hausdorff.

Theorem 2 - Let $X$ be a locally compact Hausdorff space (http://planetmath.org/LocallyCompactHausdorffSpace) and $Y\subseteq X$ a subspace. If $Y$ is locally compact then $Y$ is locally closed in $X$.