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 $X$.

Title	local compactness is hereditary for locally closed subspaces
Canonical name	LocalCompactnessIsHereditaryForLocallyClosedSubspaces
Date of creation	2013-03-22 17:36:33
Last modified on	2013-03-22 17:36:33
Owner	asteroid (17536)
Last modified by	asteroid (17536)
Numerical id	5
Author	asteroid (17536)
Entry type	Theorem
Classification	msc 54D45
Related topic	LocallyCompactHausdorffSpace