Definition:Locally Compact Space

Definition
Let $T = \left({S, \tau}\right)$ be a topological space.

Then $T$ is locally compact :
 * every point of $S$ has a neighborhood basis $\mathcal B$ such that all elements of $\mathcal B$ are compact.

Also defined as
A locally compact space is sometimes defined as what on is called a weakly locally compact space.

There is no clear consensus on the definition of local compactness, allthough there appears to be a modern trend in favor of the definition used here.

A concept that is more often used and about which there is no disagreement, is that of a locally compact Hausdorff space.

Also see

 * Definition:Locally Compact Hausdorff Space
 * Definition:Weakly Locally Compact Space
 * Definition:Strongly Locally Compact Space
 * Definition:$\sigma$-Locally Compact Space
 * Definition:Weakly $\sigma$-Locally Compact Space
 * Sequence of Implications of Local Compactness Properties