A space X is said to be locally compact (briefly L-Compact) at  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  is L-compact, since for each , we have . Thus  is a neighbourhood of x which is compact by Heine-Boral theorem. This proves that  is L-compact. But recall that  is not compact.
•  and  as subspace of  are not locally compact.

Theorems:

• A compact space is L-compact.
• If X is a Hausdorff locally compact space, then for all  and for all neighbourhoods U of x, there exists a compact neighbourhood V of x such that .
• Let  be an open continuous surjection. If X is L-compact, the Y is L-compact.
• Local compactness is a closed hereditary property.
•  are L-compact if and only if  is L-compact.