G-Delta Set is not necessarily Open Set

Theorem

Let $T = \struct {S, \tau}$ be a topological space.

Let $X$ be a $G_\delta$ set of $T$.

Then it is not necessarily the case that $X$ is a open set of $T$.

Proof

Let $T = \struct {S, \tau}$ be a finite complement topology on an uncountable set $S$.

Let $X \subseteq S$ be a $G_\delta$ set of $T$.

From $F_\sigma$ and $G_\delta$ Subsets of Uncountable Finite Complement Space:

$X$ is neither open nor closed in $T$.

Hence the result.

$\blacksquare$

Sources

• 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text I$: Basic Definitions: Section $1$: General Introduction