Closure of Subset of Indiscrete Space

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Theorem

Let $T = \left({S, \left\{{\varnothing, S}\right\}}\right)$ be an indiscrete topological space.

Let $\varnothing \subsetneq H \subseteq S$ (that is, let $H$ be a non-empty subset of $T$).


Then:

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

where:

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


Proof

From Limit Points of Indiscrete Space, every point in $S$ is a limit point of $H$.

So from the definition of closure, $H \ne \varnothing \implies H^-= S$.

From the open set axioms, $S$ is open in $T$.

From Interior of Open Set, $S^\circ = S$.

From Underlying Set of Topological Space is Clopen, $S$ is closed in $T$.

From Closed Set Equals its Closure, $S^- = S$.

The result follows.

$\blacksquare$


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