Definition:Locally Compact Hausdorff Space
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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).