Locally Compact Space is Weakly Locally Compact

Theorem
Let $T$ be a locally compact topological space.

Then $T$ is weakly locally compact.

Proof
Because $T$ is locally compact, every point of $T$ has a neighborhood basis consisting of compact sets.

By Neighborhood Basis is Non-Empty, each point has a compact neighborhood.

Thus $T$ is weakly locally compact.

Also see

 * Strongly Locally Compact Space is Weakly Locally Compact
 * Sequence of Implications of Local Compactness Properties