Every closed subset of $\mathbb{R}$ is a zero set and a $G_\delta$, as the perfect-normality criterion predicts

$\mathbb{R}$ with its usual topology is metrizable (The absolute value makes $\mathbb{R}$ a metric space: $d(x,y) = |x-y|$ is a metric, its open balls are the intervals $(x-r, x+r)$, and it is unbounded, Metrizable space: a topological space whose topology is induced by some metric; metrizability is topological, the metric is not), hence perfectly normal by In a metric space every closed set is a zero set and a $G_\delta$, and the distance function separates a point from a closed set, so every metrizable space is Tychonoff and perfectly normal: every closed $C \subseteq \mathbb{R}$ is a zero set (Zero sets and cozero sets of continuous real-valued functions) and a $G_\delta$ ($G_\delta$ and $F_\sigma$ subsets of a topological space, agreeing with the real-line notion). Taking $C := \{0\}$ makes both witnesses explicit: $C = Z(f)$ for $f(x) := |x|$, and $C = \bigcap_{n \in \mathbb{N}} \big(-\tfrac{1}{n+1},\ \tfrac{1}{n+1}\big)$.

This is exactly what Under dependent choice a space is perfectly normal if and only if it is normal and every closed set is a zero set predicts of a perfectly normal space, illustrated by the metric case that theorem's own proof does not need to run through, since perfect normality of $\mathbb{R}$ is already established directly from the metric.

• No general closed subset of $\mathbb{R}$ is exceptional here. The argument above uses nothing about $\{0\}$ beyond it being closed and nonempty in a metric space; the same two formulas, with $d(x,C)$ in place of $|x|$, exhibit any closed $C \subseteq \mathbb{R}$ as a zero set and a $G_\delta$, choice-free.

• This does not exercise the harder half of Under dependent choice a space is perfectly normal if and only if it is normal and every closed set is a zero set. That theorem's forward direction builds a zero set from a $G_\delta$ presentation via a countable family of Urysohn functions; here the zero set is read off directly from the metric, with no such construction and no dependent choice.