Interior of Closed Set of Particular Point Space

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Theorem

Let $T = \left({S, \tau_p}\right)$ be a particular point space.

Let $V \subseteq S$ be closed in $T$ such that $V \ne S$.


Then:

$V^\circ = \varnothing$

where $V^\circ$ denotes the interior of $V$.


Proof

By definition:

$\forall U \in \tau_p, U \ne \varnothing: p \in U$


Thus if $V$ is closed in $T$:

$\exists U \subseteq T: V = \complement_S \left({U}\right)$

So $p \notin V$.


Hence no open set of $T$ can be a subset of $V$ unless it is $\varnothing$.

Hence the result, by definition of interior.

$\blacksquare$


Sources