Open and Closed Sets in Indiscrete Topology

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

Then $\varnothing$ and $S$ are the only subsets of $S$ which are any one of:
 * an open set


 * a closed set


 * a $F_\sigma$ set


 * a $G_\delta$ set.

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

Open Sets
A set $U$ is open in a topology $\vartheta$ if $U \in \vartheta$.

In $\vartheta = \left\{{\varnothing, S}\right\}$, the only open sets are $\varnothing$ and $S$.

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

That is, the complements of open sets.

As we have seen, in $\vartheta = \left\{{\varnothing, S}\right\}$, 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.

$F_\sigma$ Sets
An $F_\sigma$ (F-Sigma) set is a set which can be written as a countable union of closed sets of $S$.

Hence the only $F_\sigma$ sets of $T$ are made from unions of $T$ and $\varnothing$.

So $T$ and $\varnothing$ are the only $F_\sigma$ sets of $T$.

$G_\delta$ Sets
An $G_\delta$ (G-Delta) set is a set which can be written as a countable intersection of open sets of $S$.

Hence the only $G_\delta$ sets of $T$ are made from intersections of $T$ and $\varnothing$.

So $T$ and $\varnothing$ are the only $G_\delta$ sets of $T$.