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: $d(x,y)=\mathrm{\mid }x-y\mathrm{\mid }$ 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 $f(x):=\mathrm{\mid }x\mathrm{\mid }$, and $C={\bigcap }_{n\in \mathbb{N}}(-\frac{1}{n+1},\frac{1}{n+1})$.

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 $\mathrm{\mid }x\mathrm{\mid }$, 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.