Open Sets in Indiscrete Topology

Theorem

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

Let $H \subseteq S$.

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

Proof

A set $U$ is open in a topology $\tau$ if $U \in \tau$.

In $\tau = \set {\O, S}$, the only open sets are $\O$ and $S$.

$\blacksquare$

Sources

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