# Closed Sets in Indiscrete Topology

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## Theorem

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

Let $H \subseteq S$.

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

## Proof

A set $U$ is closed in a topology $\tau$ if $\complement_S \left({U}\right) \in \tau$, where $\complement_S \left({U}\right)$ denotes the complement of $U$ in $S$.

That is, the complements of open sets.

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

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

- $\complement_S \left({\varnothing}\right) = S$ from Relative Complement of Empty Set

and:

- $\complement_S \left({S}\right) = \varnothing$ from Relative Complement with Self is Empty Set

as stated.

$\blacksquare$

## Sources

- 1970: Lynn Arthur Steen and J. Arthur Seebach, Jr.:
*Counterexamples in Topology*... (previous) ... (next): $\text{II}: \ 4: \ 2$