Let $\mathcal{X}$ be a collection of extended metric spaces. Let $\Pi\mathcal{X} = \Pi_{X \in \mathcal{X}} X$. For $1 \leq p \leq \infty$, define the following extended metric on $\Pi\mathcal{X}$:
$d_p: \Pi\mathcal{X} \times \Pi\mathcal{X} \to [0, \infty], d_p((x_X)_{X \in \mathcal{X}}, (y_X)_{X \in \mathcal{X}}) = \|(d_X(x_X, y_X))_{X \in \mathcal{X}}\|_p$
In particular, for $p = \infty$:
$d_\infty: \Pi\mathcal{X} \times \Pi\mathcal{X} \to [0, \infty], d_\infty((x_X)_{X \in \mathcal{X}}, (y_X)_{X \in \mathcal{X}}) = \underset{X \in \mathcal{X}}{\operatorname{sup}} d_X(x_X, y_X)$
This turns $(\Pi\mathcal{X}, d_p)$ into an extended metric space.
Is this construction well-known? Does it have a common name? I would appreciate any reference.