Closed Subset of Real Number Line is G-Delta

Theorem

Let $\struct {\R, \tau_d}$ be the real number line with the usual (Euclidean) topology.

Let $H \subseteq \R$ be a closed subset of $\R$.

Then $H$ is a $G_\delta$ set.

Proof

We have:

Real Number Line is Metric Space

Closed Set in Metric Space is $G_\delta$

Hence the result.

$\blacksquare$

Sources

• 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text {II}$: Counterexamples: $28$. Euclidean Topology: $5$