A space is said to be locally compact (briefly Compact) at if and only if has a compact neighbourhood in . If is compact at every point, then is called a locally compact space.
• Compact spaces are compact. Suppose is compact; is a neighborhood of each of its points implies is compact.
• The usual real line is compact, since for each , we have . Thus is a neighbourhood of which is compact by the Heine-Boral theorem. This proves that is compact. But recall that is not compact.
• and as a subspace of are not locally compact.
• A compact space is compact.
• If is a Hausdorff locally compact space, then for all and for all neighbourhoods of , there exists a compact neighbourhood of such that .
• Let be an open continuous surjection. If is compact, the is compact.
• Local compactness is a closed hereditary property.
• are compact if and only if is compact.