Definition:Locally Compact Hausdorff Space

Definition

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

Definition 1

$T$ is a locally compact Hausdorff space if and only if each point of $S$ has a compact neighborhood.

That is, if and only if $T$ is weakly locally compact.

Definition 2

$T$ is a locally compact Hausdorff space if and only if each point has a neighborhood basis consisting of compact sets.

That is, if and only if $T$ is locally compact (in the general sense).

Also see

• Equivalence of Definitions of Locally Compact Hausdorff Space

• For arbitrary topological spaces:
  • Definition:Locally Compact Space
  • Definition:Weakly Locally Compact Space