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.