# Closed Sets in Indiscrete Topology

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## Theorem

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

Let $H \subseteq S$.

$H$ is a closed set of $T$ if and only if either $H = S$ or $H = \O$.

## Proof

A set $U$ is closed in a topology $\tau$ if and only if:

- $\relcomp S U \in \tau$

where $\relcomp S U$ denotes the complement of $U$ in $S$.

That is, the closed sets are the complements of the open sets.

From Open Sets in Indiscrete Topology, in $\tau = \set {\O, S}$, the only open sets are $\O$ and $S$.

Hence the only closed sets in the indiscrete topology on $S$ are:

- $\relcomp S \O = S$ from Relative Complement of Empty Set

and:

- $\relcomp S S = \O$ from Relative Complement with Self is Empty Set

as stated.

$\blacksquare$

## Sources

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