## Uniform spaces

Let $X$ be a set and let $N$ be a subset of $X×X$. Define $NΔ = {x | (x,x)∈N} , Nt = {(y,x) | (x,y)∈N},$ and $N×XN = { (x,y) | there exists z∈X such that (x,z) ∈M and (z,y) ∈M }.$

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

(a)   If $N\subseteq X×X$, $U\in 𝒳$ 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 $\left\{\left(x,x\right)\phantom{\rule{0.2em}{0ex}}|\phantom{\rule{0.2em}{0ex}}x\in X\right\}\subseteq N$.
(d)   If $N\in 𝒳$ then $\left\{\left(x,y\right)\phantom{\rule{0.2em}{0ex}}|\phantom{\rule{0.2em}{0ex}}\left(y,x\right)\in N\right\}\in 𝒳$.
(e)   If $N\in 𝒳$ then there exists $M\in 𝒳$ such that $M{×}_{X}M\subseteq N$.
Uniformly continuous functions are for comparing uniform spaces.
Let $\left(X,𝒳\right)$ and $\left(Y,𝒴\right)$ 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 $\left(f\left(x\right),f\left(y\right)\right)\in W$.

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

 ${B}_{V}\left(x\right)=\left\{y\in X\phantom{\rule{.5em}{0ex}}|\phantom{\rule{.5em}{0ex}}\left(x,y\right)\in V\right\}$.
HW: Let $f:X\to Y$ be a uniformly continuous function. Then $f:X\to Y$ is continuous.

## Cauchy filters and completions

Let $\left(X,𝒳\right)$ 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 $N×N\subseteq D$.

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 $\stackrel{ˆ}{X}$ is constructed as $Xˆ = { minimal Cauchy filters x^ on X}$ with uniform structure $\stackrel{^}{𝒳}$ generated by the sets $Vˆ = { (x^,y^) | there exists N∈x^ ∩y^ such that N×N⊆V },$ for $V\in 𝒳$ such that if $\left(x,y\right)\in V$ then $\left(y,x\right)\in V$ and uniformly continuous map $ι:X→ Xˆ given by \iota \left(x\right)=𝒩\left(x\right), where 𝒩\left(x\right) is the neighborhood filter of x.$

HW: Show that $\stackrel{ˆ}{X}$ is a universal object in the cateogry of Hausdorff uniform spaces: Let $\left(X,𝒳\right)$ be a uniform space. The completion of $X$ is a complete Hausdorff uniform space $\stackrel{ˆ}{X}$ with a uniformly continuous map $\iota :X\to \stackrel{ˆ}{X}$ 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 $g:\stackrel{ˆ}{X}\to Y$ such that $f=g\circ \iota$.

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## Metric spaces

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

(a)   If $x\in X$ then $d\left(x,x\right)=0$,
(b)   If $x,y\in X$ and $d\left(x,y\right)=0$, then $x=y$,
(c)   If $x,y,z\in X$ then $d\left(x,z\right)\le d\left(x,y\right)+d\left(y,z\right)$.

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

 ${B}_{\epsilon }\left(x\right)=\left\{p\in X\phantom{\rule{0.2em}{0ex}}|\phantom{\rule{0.2em}{0ex}}d\left(x,y\right)\le \epsilon \right\}$.
Let $X$ be a metric space. The metric space uniformity on $X$ is the uniformity consisting of the sets $Dε = xy | dxy <ε for ε∈ ℝ>0$ The metric space topology on $X$ is the topology generated by the sets
 ${B}_{\epsilon }\left(x\right),\phantom{\rule{2em}{0ex}}$ for $x\in X$ and $\epsilon \in {ℝ}_{>0}$.

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

 $\stackrel{^}{X}=\left\{\text{Cauchy sequences}\phantom{\rule{0.2em}{0ex}}\stackrel{^}{x}\phantom{\rule{0.2em}{0ex}}\text{in}\phantom{\rule{0.2em}{0ex}}X\right\}/\sim$
with metric $d:\stackrel{^}{X}×\stackrel{^}{X}\to {ℝ}_{\ge 0}$ defined by
 $d\left(\stackrel{^}{x},\stackrel{^}{y}\right)=\underset{n\to \infty }{\mathrm{lim}}d\left({x}_{n},{y}_{n}\right)$.
where two Cauchy sequences $\stackrel{^}{x}$ and $\stackrel{^}{y}$ are equivalent,
 $\stackrel{^}{x}\sim \stackrel{^}{y}$,     if    $\underset{n\to \infty }{\mathrm{lim}}d\left({x}_{n},{y}_{n}\right)=0$.

## Examples

(a)   completion of $ℤ$ to get $ℤ$.
(a)   completion of $ℚ$ to get $ℝ$.
(b)   completion of $ℤ$ to get ${ℤ}_{p}$.
(b)   completion of $ℚ$ to get ${ℚ}_{p}$.
(c)   completion of $𝔽\left[x\right]$ to get $F\left[\left[x\right]\right]$.
(c)   completion of $𝔽\left(x\right)$ to get $F\left(\left(x\right)\right)$.
(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.