Compact Space in Particular Point Space

Theorem

Let $T = \struct {S, \tau_p}$ be a particular point space.


Then $\set p$ is compact in $T$.


Proof

Any open cover of $\set p$ has a finite subcover: any single set that contains $p$ is a cover for $\set p$.

So $\set p$ is compact in $T$.

$\blacksquare$


Also see


Sources

  • 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text {II}$: Counterexamples: $8 \text { - } 10$. Particular Point Topology: $5$
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