On $C(\mathbb{R}, \mathbb{R})$ the compact-open topology has the sets $\{g : \sup_{[-m,m]} |f-g| < \varepsilon\}$ as a neighbourhood base, and $\mathbb{R}$ is locally compact so evaluation is continuous

Let $\mathbb{R}$ carry its usual metric $d(s,t) = |s-t|$ (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) and let $C(\mathbb{R},\mathbb{R})$ carry the compact-open topology (The compact-open topology on $C(X,Y)$ for a metric domain $X$, with subbasis $S(K,V) = \{f : f[K] \subseteq V\}$). For a natural $m \ge 1$ write $[-m,m] := \{\, t \in \mathbb{R} : -\iota(m) \le t \le \iota(m) \,\}$ (Intervals of $\mathbb{R}$: the nine order-convex forms, nondegeneracy, and length, The canonical natural $\iota(n) = n \cdot 1_F$ of a field). Then:

1. every $[-m,m]$ is a compact subset of $\mathbb{R}$, and every compact $K \subseteq \mathbb{R}$ is contained in some $[-m,m]$;
2. for each $f \in C(\mathbb{R},\mathbb{R})$ the sets $B_{[-m,m]}(f,\varepsilon) = \{\, g \in C(\mathbb{R},\mathbb{R}) : |f(t)-g(t)| < \varepsilon \text{ for every } t \in [-m,m] \,\} \qquad (m \ge 1,\ \varepsilon > 0)$ form a neighbourhood base at $f$ in the compact-open topology (Neighbourhood of a point and neighbourhood base, with this library's convention that a neighbourhood need not be open);
3. $\mathbb{R}$ is a locally compact metric space (Locally compact metric space: every point has a compact neighbourhood), so the evaluation map $e : C(\mathbb{R},\mathbb{R}) \times \mathbb{R} \to \mathbb{R}$ is continuous (If $X$ is a locally compact metric space then the evaluation map is continuous for the compact-open topology, The evaluation map $e : C(X,Y) \times X \to Y$, $e(f,x) = f(x)$).

The quantity $\sup_{t \in [-m,m]} |f(t)-g(t)|$ of the title exists and is a maximum, by fact (U3) of The topology of compact convergence on $C(X,Y)$ for metric $X$ and $Y$: uniform convergence on each compact subset of $X$; the formulation in claim 2 avoids writing it, which is what keeps the empty compact set harmless elsewhere on this page.

• Claim 2 is what makes the compact-open topology on $C(\mathbb{R},\mathbb{R})$ concrete. A general neighbourhood in it involves an arbitrary compact set and an arbitrary open subset of the target; claim 2 replaces both by a bound on a symmetric interval and a single $\varepsilon$, and the intervals may be indexed by the naturals. That is the shape a metrization proof would exploit, and this library does not carry out that proof.

• Local compactness of $\mathbb{R}$ is where Heine-Borel is spent. In a general metric space a closed ball need not be compact, and then nothing above survives; what makes $\mathbb{R}$ work is that closed bounded sets are compact (Heine-Borel in $\mathbb{R}^n$: with the Euclidean metric a subset of $\mathbb{R}^n$ is compact if and only if it is closed and bounded, and the proof by bisection uses no choice principle; the same holds on the real line). The contrast is $\mathbb{Q}$, where the evaluation map is not continuous at all.

• The intervals $[-m,m]$ exhaust $\mathbb{R}$, and that is claim 1's real content. Every compact subset sits inside one of countably many of them, so the compact sets, of which there are very many, are controlled by a countable family. Nothing about metrizability follows from this alone, and none is claimed.