# Closure of Subset of Indiscrete Space

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## Theorem

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

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

Then:

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

where:

## Proof

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

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

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.

$\blacksquare$

## Sources

- 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.:
*Counterexamples in Topology*(2nd ed.) ... (previous) ... (next): Part $\text {II}$: Counterexamples: $4$. Indiscrete Topology: $6$