Open Set is G-Delta Set

Theorem

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

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


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


Proof

$U$ is the intersection of a singleton.

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

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

$\blacksquare$


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
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