# Open and Closed Sets in Indiscrete Topology

## Theorem

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

Let $H \subseteq S$.

### Open Sets in Indiscrete Topology

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

### Closed Sets in Indiscrete Topology

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

### $F_\sigma$ Sets in Indiscrete Topology

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

### $G_\delta$ Sets in Indiscrete Topology

$H$ is a $G_\delta$ ($G$-delta) set of $T$ if and only if either $H = S$ or $H = \O$.