Local Compact

A space X is said to be locally compact (briefly L - Compact) at x \in X if and only if x has a compact neighbourhood in X. If X is L - compact at every point, then X is called a locally compact space.

• Compact spaces are L - compact. Suppose X is compact; X is a neighborhood of each of its points implies X is L - compact.
• The usual real line \mathbb{R} is L - compact, since for each x \in \mathbb{R}, we have x \in \left( {a,b} \right) \subseteq \left[ {a,b} \right]. Thus \left[ {a,b} \right] is a neighbourhood of x which is compact by the Heine-Boral theorem. This proves that \mathbb{R} is L - compact. But recall that \mathbb{R} is not compact.
\mathbb{Q} and {\mathbb{Q}^c} as a subspace of \mathbb{R} are not locally compact.

• A compact space is L - compact.
• If X is a Hausdorff locally compact space, then for all x \in X and for all neighbourhoods U of x, there exists a compact neighbourhood V of x such that V \subseteq U.
• Let f:X \to Y be an open continuous surjection. If X is L - compact, the Y is L - compact.
• Local compactness is a closed hereditary property.
{X_1},{\text{ }}{X_2} are L - compact if and only if {X_1} \times {X_2} is L - compact.