Interior of Subset of Indiscrete Space

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Theorem

Let $T = \struct {S, \set {\O, S} }$ be an indiscrete topological space.

Let $H \subset S$ (that is, let $H$ be a proper subset of $T$).


Then:

$H^\circ = H^{\circ -} = H^{\circ - \circ} = \O$

where:

$H^\circ$ denotes the interior of $H$
$H^-$ denotes the closure of $H$.


Proof

As $H \subset S$, it follows that $H \ne T$.

So the only open subset of $H$ is $\O$.

So by definition:

$H^\circ = \O$

From Empty Set is Closed in Topological Space, $\O$ is closed in $T$.

From Closed Set Equals its Closure:

$\O^- = \O$

From Empty Set is Element of Topology, $\O$ is open in $T$.

From Interior of Open Set:

$\O^\circ = \O$

The result follows.

$\blacksquare$


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