G-Delta Set is not necessarily Open Set
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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:
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