Closure of Open Set of Particular Point Space

Theorem

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

Let $U \in \tau_p$ be open in $T$ such that $U \ne \O$.

Then:

$U^- = S$

where $U^-$ denotes the closure of $U$.

Proof

Follows directly from:

Particular Point Topology is Closed Extension Topology of Discrete Topology

Closure of Open Set of Closed Extension Space

$\blacksquare$

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: $2$