Open Sets are G-Delta Sets

Theorem

Let $\left({X, \vartheta}\right)$ be a topological space.

Let $U$ be an open set of $X$.

Then $U$ is a $G_\delta$ set of $X$.

Proof

$U$ is the intersection of a singleton.

So $U$ is trivially the intersection of a countable number of open sets of $X$.

The result follows by definition of $G_\delta$ set.

$\blacksquare$

Sources

• Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (1970)... (previous)... (next): $\text{I}: \ \S 1$