# Interior of Closed Set of Particular Point Space

## Theorem

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

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

Then:

$V^\circ = \O$

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

## Proof

By definition:

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

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

$\exists U \subseteq T: V = \relcomp S U$

So $p \notin V$.

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

Hence the result, by definition of interior.

$\blacksquare$