I'm examining proofs made on Page 12 of this document and in Billingsley's book Convergence of Probability Measures (first edition, page 12). I'm trying to extend this to an infinite product space with arbitrary metrics.
Let $\pi_k : \mathbb{R}^{\infty} \to \mathbb{R}^k$ be the usual mapping (the first $k$ coordinates of $x = (x_1, x_2, \ldots) \in \mathbb{R}^\infty$); the class of finite dimensional subsets of $\mathbb{R}^{\infty}$ are sets of the form $\pi_k^{-1} H$ with $H \subset \mathbb{R}^k$. Both references I mentioned above argue that this class of sets is a convergence-determining class. Billingsley's book uses Corollary 1 from Section 2 of Chapter 1 to make this point; the linked reference uses its Theorem 2.4. In either case, to invoke the corollary/theorem, you need that for $x \in \mathbb{R}^\infty$ and an arbitrary $\epsilon > 0$, there must be an $A$ in the class of sets in question such that $x \in A^\circ \subset A \subset B(x, \epsilon)$, with $B(x, r)$ being the open ball centered at $x$ of radius $r$ in $\mathbb{R}^\infty$ (the metric on $\mathbb{R}^\infty$ is $\rho(x, y) = \sup_{i \geq 1} \left|x_i - y_i\right|$) and $A^\circ$ the interior of $A$.
Neither reference shows that this is the case, and not only that it seems that this is not the case. For instance, both references work with sets $N_{k, \epsilon}(x) = \{y: \max_{1 \leq i \leq k} \left|x_i - y_i \right| < \epsilon\}$, but this is not a subset of $B(x, \epsilon)$; in fact, $B(x, \epsilon) \subset N_{k, \epsilon}(x)$.
Both references allude to the fact that if $\delta < \epsilon$, $\partial N_{k, \delta}(x) \cap \partial N_{k, \epsilon}(x) = \emptyset$, which I don't see as important.
Can someone explain these issues?