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.

$\blacksquare$

Also see

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