Infinite Particular Point Space is not Strongly Locally Compact
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Theorem
Let $T = \struct {S, \tau_p}$ be an infinite particular point space.
Then $T$ is not strongly locally compact.
Proof
By definition, $T$ is strongly locally compact if and only if every point of $S$ is contained in an open set whose closure is compact.
Let $x \in S: x \ne p$.
Let $x \in U$ where $U$ is open in $T$.
From Closure of Open Set of Particular Point Space we have:
- $U^- = S$
where $U^-$ is the closure of $U$.
But from Infinite Particular Point Space is not Compact we have that $S$ is not compact.
Hence the result, by definition of strongly locally compact.
$\blacksquare$
Sources
- 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text {II}$: Counterexamples: $9 \text { - } 10$. Infinite Particular Point Topology: $5$
- 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text {II}$: Counterexamples: $9 \text { - } 10$. Infinite Particular Point Topology: $15$