Existence of Weakly Sigma-Locally Compact Space which is not Strongly Locally Compact
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Theorem
There exists at least one example of a weakly $\sigma$-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 $\sigma$-Locally Compact, $T$ is a weakly $\sigma$-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