Uniform spaces, metric spaces and completion

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au

Last updates: 29 April 2011

Uniform spaces

Let $X$ be a set and let $N$ be a subset of $X\times X$. Define $$N^{\Delta }=\left\{x\right|(x,x)\in N\},N^t=\{(y,x)\left|\right(x,y)\in N\},$$ and $$N{\times }_{X}N=\left\{\right(x,y\left)\right|\text{there exists}z\in X\text{such that}(x,z)\in M\text{and}(z,y)\in M\}.$$

A uniform space is a set $X$ with a collection $𝒳$ of subsets of $X\times X$ such that

(a)   If $N\subseteq X\times X$, $U\in 𝒳$ and $N\supseteq U$ then $N\in 𝒳$.

(b)   If $N_1,\dots ,N_{\ell }\in 𝒳$ then $N_1\cap \cdots \cap N_{\ell }\in 𝒳$.

(c)   If $N\in 𝒳$ then $\left\{\right(x,x\left)\right|x\in X\}\subseteq N$.

(d)   If $N\in 𝒳$ then $\left\{\right(x,y\left)\right|(y,x)\in N\}\in 𝒳$.

(e)   If $N\in 𝒳$ then there exists $M\in 𝒳$ such that $M{\times }_{X}M\subseteq N$.

Uniformly continuous functions are for comparing uniform spaces.
Let $(X,𝒳)$ and $(Y,𝒴)$ be uniform spaces. A uniformly continuous function from $X$ to $Y$ is a function $f:X\to Y$ such that

if $W\in 𝒴$ then there exists $V\in 𝒳$ such that if $\left(x,y\right)\in V$ then $\left(f\right(x),f(y\left)\right)\in W$.

Let $(X,𝒳)$ be a uniform space. The uniform space topology on $X$ is the topology generated by the sets

$B_V\left(x\right)=\{y\in X|(x,y)\in V\}$.

HW: Let $f:X\to Y$ be a uniformly continuous function. Then $f:X\to Y$ is continuous.

Cauchy filters and completions

Let $(X,𝒳)$ be a uniform space. A Cauchy filter is a filter $ℱ$ on $X$ such that

if $D\in 𝒳$ then there exists $N\in ℱ$ such that $N\times N\subseteq D$.

HW: A convergent filter is Cauchy.

A complete space is a uniform space for which every Cauchy filter converges.

A minimal Cauchy filter is a Cauchy filter which is minimal with respect to inclusion of filters. The completion of $X$ is the uniform space $\hat{X}$ is constructed as $$\hat{X}=\left\{\text{minimal Cauchy filters}\hat{x}\text{on}X\right\}$$ with uniform structure $\widehat{𝒳}$ generated by the sets $$\hat{V}=\left\{\right(\hat{x},\hat{y}\left)\right|\text{there exists}N\in \hat{x}\cap \hat{y}\text{such that}N\times N\subseteq V\},$$ for $V\in 𝒳$ such that if $(x,y)\in V$ then $(y,x)\in V$ and uniformly continuous map $$\iota :X\to \hat{X}\text{given by}$ \iota \left(x\right)=𝒩\left(x\right),$where$ 𝒩\left(x\right)$is\; the\; neighborhood\; filter\; of$ x$.$$

HW: Show that $\hat{X}$ is a universal object in the cateogry of Hausdorff uniform spaces: Let $(X,𝒳)$ be a uniform space. The completion of $X$ is a complete Hausdorff uniform space $\hat{X}$ with a uniformly continuous map $\iota :X\to \hat{X}$ such that if $Y$ is a complete Hausdorff uniform space and $f:X\to Y$ is a uniformly continuous map then there exists a unique $g:\hat{X}\to Y$ such that $f=g\circ \iota$.

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Metric spaces

A metric space is a set $X$ with a function $d:X\times X\to {ℝ}_{\ge 0}$ such that

(a)   If $x\in X$ then $d(x,x)=0$,

(b)   If $x,y\in X$ and $d(x,y)=0$, then $x=y$,

(c)   If $x,y,z\in X$ then $d(x,z)\le d(x,y)+d(y,z)$.

Let $X$ be a metric space. Let $x\in X$ and let $\epsilon \in {ℝ}_{>0}$. The ball of radius $\epsilon$ at $x$ is the set

$B_{\epsilon }\left(x\right)=\{p\in X\left|d\right(x,y)\le \epsilon \}$.

Let $X$ be a metric space. The metric space uniformity on $X$ is the uniformity consisting of the sets $$D_{\epsilon }=\left\{\left(x,y\right)|d\left(x,y\right)<\epsilon \right\}\text{for}\epsilon \in {ℝ}_{>0}$$ The metric space topology on $X$ is the topology generated by the sets

$B_{\epsilon }\left(x\right),$ for $x\in X$ and $\epsilon \in {ℝ}_{>0}$.

Let $X$ be a metric space. The completion of $X$ is the metric space

$\hat{X}=\left\{\text{Cauchy sequences}\hat{x}\text{in}X\right\}/\sim$

with metric $d:\hat{X}\times \hat{X}\to {ℝ}_{\ge 0}$ defined by

$d(\hat{x},\hat{y})=\underset{n\to \infty }{\lim }d(x_n,y_n)$.

where two Cauchy sequences $\hat{x}$ and $\hat{y}$ are equivalent,

$\hat{x}\sim \hat{y}$,     if    $\underset{n\to \infty }{\lim }d(x_n,y_n)=0$.

Trying to make a category of metric spaces: Lipschitz continuity, isometries

Examples

(a)   completion of $\mathbb{Z}$ to get $\mathbb{Z}$.

(a)   completion of $\mathbb{Q}$ to get $ℝ$.

(b)   completion of $\mathbb{Z}$ to get ${\mathbb{Z}}_p$.

(b)   completion of $\mathbb{Q}$ to get ${\mathbb{Q}}_p$.

(c)   completion of $𝔽\left[x\right]$ to get $F\left[\right[x\left]\right]$.

(c)   completion of $𝔽\left(x\right)$ to get $F\left(\right(x\left)\right)$.

(d)   completion of $C_c\left(X\right)$ to get $C_0\left(X\right)$.

(d)   completion of $C_c\left(X\right)$ to get $L^p\left(\mu \right)$.

Notes and References

These notes follow Bourbaki [Bou] Chapter II. The category of uniform spaces is a natural home for uniformly continuous functions, Cauchy sequences and completion. The treatment of metric spaces and completion follows [BR] Chapter 2 Exercise 24. A uniform space is almost a metric space: By [Bou??] the separable Hausdorff uniform spaces are exactly the separable metric spaces.

References

[Bou] N. Bourbaki, General Topology, Springer-Verlag, 1989. MR1726779.

[BR] W. Rudin, Principles of mathematical analysis, Third edition, International Series in Pure and Applied Mathematics, McGraw-Hill 1976. MR0385023.

[Ru] W. Rudin, Real and complex analysis, Third edition, McGraw-Hill, 1987. MR0924157.

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