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.

Examples:
• Compact spaces are $L -$ compact. Suppose $X$ is compact, $X$ is a neighbourhood 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 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 subspace of $\mathbb{R}$ are not locally compact.

Theorems:
• 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.