Definition:Locally Compact Space

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

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

Alternative Definition
There is an alternative definition which is often seen:


 * $T$ is locally compact iff every point of $S$ is contained in a compact neighborhood.

However, this definition is equivalent to the main definition only when $T$ is a $T_2$ (Hausdorff) space.

Note that if there exists a local basis of compact sets, there is a compact neighborhood (any of the sets that form the local basis).

Thus the main definition trivially implies the alternative one.

Also see

 * Local Compactness in Hausdorff Space