Existence of Weakly Locally Compact Space which is not Strongly Locally Compact/Proof 1

Theorem

There exists at least one example of a weakly locally compact topological space which is not also a strongly locally compact space.

Proof

Let $T$ be the nested interval topological space.

From Nested Interval Topology is Weakly Locally Compact, $T$ is a weakly locally compact space.

From Nested Interval Topology is not Strongly Locally Compact, $T$ is not a strongly locally compact space.

Hence the result.

$\blacksquare$

Sources

• 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text I$: Basic Definitions: Section $3$: Compactness: Localized Compactness Properties