# Local Compact

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.

**Examples:**

• Compact spaces are compact. Suppose is compact, is a neighbourhood 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 Heine-Boral theorem. This proves that is compact. But recall that is not compact.

• and as subspace of are not locally compact.

**Theorems:**

• 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.