Borel sigma-algebra of continuous path space is generated by coordinates

Fix $t\ge 0$ and choose an integer $n\ge 1$ with $t\le n$. If $\epsilon >0$, then $d(f,g)<2^{-n}\min \{1,\epsilon \}$ implies $2^{-n}(1\wedge M_n(f,g))<2^{-n}\min \{1,\epsilon \}$ and hence $\mathrm{\mid }{\pi }_t(f)-{\pi }_t(g)\mathrm{\mid }\le M_n(f,g)<\epsilon$. Thus every ${\pi }_t$ is continuous, so it is Borel measurable. Minimality in [F2] gives ${\mathcal{G}}_{\mathbb{Q}}\subseteq \mathcal{G}\subseteq \mathcal{B}(C)$.

Fix $t\ge 0$ and put $q_j=2^{-j}\lfloor 2^{j}t\rfloor$. Then $q_j\in \mathbb{Q}\cap [0,\mathrm{\infty })$ and $0\le t-q_j<2^{-j}$, so $q_j\to t$ by [F6]. For every $f\in C$, continuity gives ${\pi }_{q_j}(f)\to {\pi }_t(f)$. Every ${\pi }_{q_j}$ is ${\mathcal{G}}_{\mathbb{Q}}$-measurable by definition, so [F3] makes ${\pi }_t$ measurable. Hence $\mathcal{G}\subseteq {\mathcal{G}}_{\mathbb{Q}}$ by minimality, and therefore $\mathcal{G}={\mathcal{G}}_{\mathbb{Q}}$.

Let $D$ be the family of paths which, for some integers $N,m\ge 1$, are affine on each interval $[k\mathrm{/}m,(k+1)\mathrm{/}m]$ for $0\le k<mN$, take rational values at all grid points $k\mathrm{/}m$, and are constant after $N$. This family is countable without choice. Indeed, fix bijections $e:\mathbb{N}\to \mathbb{Q}$ and $p:\mathbb{N}\to {\mathbb{N}}^2$. Nested use of $p^{-1}$ codes every finite sequence of naturals by one natural (and repeated application of $p$ decodes it); composing entries with $e$ codes every finite rational sequence. A further finite nesting codes $(N,m)$ together with that sequence, giving a surjection from a subset of $\mathbb{N}$ onto $D$. Thus [F4] makes $D$ at most countable.

The family $D$ is uoc dense. Given $f\in C$ and $\epsilon >0$, use [F6] to choose $N$ with ${\sum }_{n>N}{2}^{-n}<\epsilon \mathrm{/}2$. By [F5], $f$ is uniformly continuous on $[0,N]$; choose $m$ so that the oscillation of $f$ over distances at most $1\mathrm{/}m$ is less than $\epsilon \mathrm{/}4$. By rational density, choose rational $a_k$ with $\mathrm{\mid }{a}_k-f(k\mathrm{/}m)\mathrm{\mid }<\epsilon \mathrm{/}4$ for the finitely many $0\le k\le mN$ (finite induction, not a choice principle), and let $g\in D$ interpolate these values and remain constant after $N$. On a grid interval, convex interpolation and the triangle inequality give $\mathrm{\mid }g(t)-f(t)\mathrm{\mid }<\epsilon \mathrm{/}2$. Consequently $d(f,g)\le {\sum }_{n=1}^N{2}^{-n}{M}_n(f,g)+{\sum }_{n>N}{2}^{-n}<\epsilon \mathrm{.}$ This also covers $t=0$ and $t=N$.

Fix $f\in C$. For $n\ge 1$, continuity and rational density give $M_n(g,f)={\sup }_{q\in \mathbb{Q}\cap [0,n]}\mathrm{\mid }{\pi }_q(g)-f(q)\mathrm{\mid }\mathrm{.}$ Enumerating that rational set, [F3] makes $M_n(\cdot ,f)$ ${\mathcal{G}}_{\mathbb{Q}}$-measurable. The finite partial sums of the metric formula are measurable by [F3] and converge pointwise to $d(\cdot ,f)$, so this distance is measurable. Hence every metric ball, with an arbitrary center, belongs to ${\mathcal{G}}_{\mathbb{Q}}$.

The balls $B(h,r)$ with $h\in D$ and positive rational $r$ form a countable base: countability follows from [F4], while density from step 1.4 and the triangle inequality put such a ball around every point inside any prescribed open ball. For an open $U$, let ${\mathcal{V}}_U$ be the subfamily of these basic balls which are contained in $U$. It is countable by [F4], every member belongs to ${\mathcal{G}}_{\mathbb{Q}}$ by step 1.5, and $U=\bigcup {\mathcal{V}}_U$. Thus every open set is in ${\mathcal{G}}_{\mathbb{Q}}$; [F2] yields $\mathcal{B}(C)\subseteq {\mathcal{G}}_{\mathbb{Q}}$. Combining this with steps 1.1 and 1.2 proves all stated equalities. The zero path shows $C$ and $D$ are nonempty; singleton and zero-time coordinates cause no exception, and no countable family of nonempty sets was selected.