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I was reading Wikipedia's definition of uniform spaces in terms of uniform covers. I wonder how a "uniform cover" of a set is defined? I just can't find it anywhere. Thanks!

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@DylanMoreland: Thanks! I don't quite understand your comment. Do you mean a uniform cover of a set is a family of covers of the set? Does "it's part of the data you need to give" mean the definition varies for different structures on the set? –  Tim Jan 2 '12 at 20:54
    
Actually, reading it more carefully I think they're giving the definition of a uniform cover right then and there, in the axioms (1)-(3). –  Dylan Moreland Jan 2 '12 at 20:59

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up vote 3 down vote accepted

One doesn’t define a uniform cover, but rather a family of them. The definition is actually given in the first sentence at the link that you gave, but it may need to be unpacked a bit:

A uniform space $(X,\Theta)$ is a set $X$ equipped with a distinguished family of uniform covers $\Theta$ from the set of coverings of $X$, forming a filter when ordered by star refinement.

In other words, any collection $\mathscr{C}$ of subsets of $X$ such that $\bigcup\mathscr{C}=X$ is a cover of $X$. A collection $\Theta$ of covers is a family of uniform covers of $X$ provided that it is a filter with respect to the ordering $<^*$, called star refinement, defined as follows:

A cover $\mathscr{R}$ is a star refinement of a cover $\mathscr{C}$, written $\mathscr{R}<^*\mathscr{C}$, iff for each $R\in\mathscr{R}$ there is a $C\in\mathscr{C}$ such that $\operatorname{st}(R,\mathscr{R})\subseteq C$, where $\operatorname{st}(R,\mathscr{R})=\bigcup\{S\in\mathscr{R}:S\cap R\ne\varnothing\}$. (This set is the star of $R$ with respect to $\mathscr{R}$.)

Saying that $\Theta$ is a filter with respect $<^*$ means that the following conditions hold:

  1. The trivial cover $\{X\}$ belongs to $\Theta$.
  2. If $\mathscr{R}\in\Theta$, and $\mathscr{C}$ is any cover of $X$ such that $\mathscr{R}<^*\mathscr{C}$, then $\mathscr{C}\in\Theta$.
  3. For any $\mathscr{C},\mathscr{D}\in\Theta$ there is some $\mathscr{R}\in\Theta$ such that $\mathscr{R}<^*\mathscr{C}$ and $\mathscr{R}<^*\mathscr{D}$.

The members of a family $\Theta$ of covers satisfying these conditions are called the uniform covers of the uniform space $(X,\Theta)$, but that notion makes sense only in the context of the entire family $\Theta$.

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@Tim: The notions are related. If you start with a diagonal uniformity, you can define uniform cover as in David’s answer. You then prove that the collection of them is a family of uniform covers in the sense of the Wikipedia definition. Conversely, if you start with a family of uniform covers as in the WP definition, and for each $\mathscr{C}\in\Theta$ form $D_\mathscr{C}=\bigcup\{C\times C:C\in\mathscr{C}\}$, $\{D_\mathscr{C}:\mathscr{C}\in\Theta\}$ is a diagonal uniformity whose uniform covers in David’s sense are precisely the members of $\Theta$. –  Brian M. Scott Feb 6 '12 at 7:44

I'm not sure if this is what you want, but from Willard's General Topology:

Let $\Delta$ denote the diagonal $\{(x,x):x\in X\}$ in $X\times X$.

For $U$ and $V$ subsets of $X\times X$, denote $U\circ V=\{(x,y):\text{ for some } z, (x,z)\in V\text{ and } (z,y)\in U\}$.

For $U\in X\times X$, we set $E^{-1}=\{(y,x):(x,y)\in E \}$.

A diagonal uniformity on a set $X$ is a collection ${\cal D}(X)$ of subsets of $X\times X$, called surroundings, which satisfy the conditions a)-e) below:

a) $D\in \cal D\quad\Longrightarrow\quad \Delta\subset D$.

b) $D_1, D_2\in{\cal D}\quad\Longrightarrow\quad D_1\cap D_2 \in \cal D$.

c) $D\in{\cal D}\quad\Longrightarrow\quad E\circ E\subset D$ for some $E\in\cal D$.

d) $D\in{\cal D}\quad\Longrightarrow\quad E^{-1}\subset D$ for some $E\in\cal D$.

e) $D\in{\cal D}, D\subset E\quad\Longrightarrow\quad E\in \cal D$.

When $X$ has such a structure, we say it is a uniform space.

For $x\in X$ and $D\in\cal D$, define $$ D[x]=\{y\in X:(x,y)\in D\}. $$

A cover of a uniform space $(X, \cal D)$ is a uniform cover if it is refined by a cover of the form ${\cal U}_D=\{D[x]: x\in X\}$ for some $D\in \cal D$.

In the wiki article, the term is defined it seems in the last paragraph of the Uniform cover definition section.

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+1 Thanks! The treatment in the book makes things clearer. –  Tim Jan 3 '12 at 1:12

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