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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?

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1 Answer 1

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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 $$B(F,U)=\{x\in X:x_\alpha\in U(\alpha)\text{ for each }\alpha\in F\}\;.$$

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What is "diagonal uniformity"? The link it points to does not contains words "diagonal uniformity" –  porton May 23 '12 at 16:46
    
@porton: A diagonal uniformity is a uniformity defined by entourages, rather than by uniform covers or pseudometrics. –  Brian M. Scott May 23 '12 at 19:05
    
Can this be re-formulated in terms of Frechet filters? I spend a few minuted trying to simplify this formula into a statement about Frechet filters without success and as such deem it impossible. But may it be nevertheless possible? –  porton Jun 4 '12 at 16:18
    
@porton: It seems unlikely. –  Brian M. Scott Jun 4 '12 at 20:56

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