Closure of Open Set of Particular Point Space
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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:
$\blacksquare$
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: $2$