Closure in Infinite Particular Point Space is not Compact

Theorem

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

Let $A \in \tau_p$ be open in $T$.

Let $A^-$ be the closure of $A$.

Then $A^-$ is not compact.

Proof

From Closure of Open Set of Particular Point Space, we have that $A^- = S$.

The result follows from Infinite Particular Point Space is not 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: $5$