Locally Compact Space is Weakly Locally Compact

Theorem
Let $T = \struct{S, \tau}$ be a locally compact topological space.

Then $T$ is weakly locally compact.

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

From Space is Neighborhood of all its Points, $S$ is a neighborhood of each point of $S$. By assumption, each point has a compact neighborhood contained in $S$.

Thus $T$ is weakly locally compact.

Also see

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