Constructing a metric subspace of $\prod_{\alpha\in I}X_\alpha$, where $X_\alpha$ is a metric space Suppose that for each $\alpha\in I$, $(d_\alpha,X_\alpha)$ is a metric space. (The indexing set $I$ is arbitrary.) I want to find a maximal subset $Y$ of the product space $X=\prod_{\alpha\in I}X_\alpha$ that is a metric space under the supremum metric $d(x,y)=\sup_{\alpha\in I}d_\alpha(x_\alpha,y_\alpha)$.
This is, as far as I am aware, the only "natural" metric for a product of metric spaces, but it is only a metric if $\sup_{\alpha\in I}d_\alpha(x_\alpha,y_\alpha)<\infty$ for every $x,y\in Y$, and it is not clear how to select such a subset in general. Looking online, I see an example using the space of bounded functions on a normed vector space, but on an arbitrary metric there is no "base point" from which to define the notion of "bounded". What is the standard approach to metrizing such spaces? (I am aware that, e.g. $\Bbb R^\Bbb R$ is not metrizable, but there are several "reasonable" subsets of this that are metrizable.)
 A: $\newcommand{\supp}{\operatorname{supp}}$I will assume that the spaces $X_\alpha$ are non-trivial. Suppose that $Y\subseteq X$, and fix a point $p\in Y$. For $y\in Y$ let $\supp(y)=\{\alpha\in I:y_\alpha\ne p_\alpha\}$. By a basic open set in $Y$ I mean the intersection with $Y$ of a standard product basic open set in $X$, i.e., one that restricts finitely many coordinates. The intersection of countably many basic open nbhds of $p$ restricts at most countably many coordinates, so $Y$ has no countable base at $p$ unless $\bigcup_{y\in Y}\supp(y)$ is countable. Let $S=\bigcup_{y\in Y}\supp(y)$; then $Y$ is homeomorphic to a subspace of the countable product $X_S=\prod_{\alpha\in S}X_\alpha$. If $I$ is uncountable, there seems to be no non-arbitrary way to choose a countable subset of $I$, so you’re really restricting yourself to countable $I$. 
For countable $I$ the product $X$ is always metrizable, but the function that you describe does not generate the product topology even when it’s defined. For example, if $X_n=\{0,1\}$ with the discrete metric for each $n\in\omega$, $X$ as a topological space is a Cantor set, while your function is the discrete metric on $X$.
