Definition of product of uniform spaces

Question

In Wikipedia and PlanetMath product of uniform spaces is defined as the weakest uniformity on the Cartesian product making all the projection maps uniformly continuous.

But Springer's encyclopedia has a different (supposedly equivalent) definition. The sad part is that I don't understand their notation.

Could you explain me how to understand this (Springer's) definition of product of uniform spaces?

Answer 1

The notation in the Springer definition is pretty appalling:

The product of uniform spaces $(X_t,\mathfrak{A}_t),\,t\in T$, is the uniform space $(\prod X_t,\prod\mathfrak{A}_t)$, where $\prod\mathfrak{A}_t$ is the uniformity on $\prod X_t$ with as base for the entourages sets of the form $$\Big\{\big(\{x_t\},\{t_t\}\big):(x_{t_i},y_{t_i})\in V_{t_i},i=1,\dots,n\Big\},t_i\in T,V_{t_i}\in\mathfrak{A}_{t_i},n=1,2,\dots\;.$$

Here’s what I consider a somewhat more comprehensible version of this.

Suppose that $\langle X_\alpha,\mathscr{U}_\alpha\rangle$ is a uniform space for each $\alpha\in A$, where each $\mathscr{U}_\alpha$ is a diagonal uniformity. Let $X=\prod\limits_{\alpha\in A}X_\alpha$; points of $X$ have the form $x=\langle x_\alpha:\alpha\in A\rangle$. For each finite, non-empty $F\subseteq A$ and each function $U:F\to\bigcup\limits_{\alpha\in F}\mathscr{U}_\alpha$ such that $U(\alpha)\in\mathscr{U}_\alpha$ for each $\alpha\in F$ let

$$V_{F,U}=\{\langle x,y\rangle\in X\times X:\langle x_\alpha,y_\alpha\rangle\in U(\alpha)\text{ for all }\alpha\in F\}\;,$$

and let $\mathscr{V}$ be the set of all such entourages $V_{F,U}$ of the diagonal in $X\times X$. Then $\mathscr{V}$ is a base for a diagonal uniformity $\mathscr{U}$ on $X$, which we call the product uniformity on $X$.

This is analogous to the usual definition of the product topology: that defines a typical basic open set by picking a finite set $F$ of indices and an open set $U(\alpha)$ in $X_\alpha$ for each $\alpha\in F$ and letting $$B(F,U)=\{x\in X:x_\alpha\in U(\alpha)\text{ for each }\alpha\in F\}\;.$$