Existence of Weakly Locally Compact Space which is not Strongly Locally Compact
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Theorem
There exists at least one example of a weakly locally compact topological space which is not also a strongly locally compact space.
Proof 1
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.
$\blacksquare$
Proof 2
Let $T$ be an infinite particular point space.
From Particular Point Space is Weakly Locally Compact, $T$ is a weakly locally compact space.
From Infinite Particular Point Space is not Strongly Locally Compact, $T$ is not a strongly locally compact space.
Hence the result.
$\blacksquare$