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

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