## Uniform spaces

Last update: 23 July 2014

## Uniform spaces

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

 (a) if $V\subseteq X×X$ and $B\in 𝒳$ and $B\subseteq V$ then $V\in 𝒳\text{,}$ (b) if $\ell \in {ℤ}_{>0}$ and ${V}_{1},{V}_{2},\dots ,{V}_{\ell }\in 𝒳$ then $V1∩V2∩⋯∩Vℓ ∈𝒳,$ (c) if $V\in 𝒳$ then $\left\{\left(x,x\right) | x\in X\right\}\subseteq V\text{,}$ (d) if $V\in 𝒳$ then $\left\{\left(y,x\right) | \left(x,y\right)\in V\right\}\in 𝒳,$ (e) if $V\in 𝒳$ then there exists $M\in 𝒳$ such that $M{×}_{X}M\subseteq V$ where $M×XM= { (x,y) | there exists z∈X such that (x,z)∈M and (z,y)∈M } .$

Let $\left(X,𝒳\right)$ be a uniform space. An entourage is a set in $𝒳\text{.}$

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 $\left(f\left(x\right),f\left(y\right)\right)\in W\text{.}$

Let $\left(X,𝒳\right)$ be a uniform space. The uniform space topology on $X$ is the topology on $X$ such that if $x\in X$ then $𝒩(x)= { BV(x)= {y∈X | (x,y)∈V} | V∈𝒳 }$ is the neighbourhood filter of $x\text{.}$

Homework: Let $\left(X,𝒳\right)$ and $\left(Y,𝒴\right)$ be uniform spaces and let $f:X\to Y$ be a uniformly continuous function. Show that $f:X\to Y$ is continuous (with respect to the uniform space topology on $X$ and $Y\text{).}$

## Cauchy filters and Cauchy sequences

Let $\left(X,𝒳\right)$ be a uniform space.

A Cauchy filter is a filter $ℱ$ on $X$ such that if $V\in 𝒳$ then there exists $N\in ℱ$ such that $N×N\subseteq V\text{.}$

A Cauchy sequence is a sequence ${x}_{1},{x}_{2},\dots$ in $X$ such that if $V\in 𝒳$ then there exists ${n}_{0}\in {ℤ}_{>0}$ such that if $m,n\in {ℤ}_{>0}$ and $m>{n}_{0}$ and $n>{n}_{0}$ then $\left({x}_{m},{x}_{n}\right)\in V\text{.}$

Homework: Let $\left(X,𝒳\right)$ be a uniform space and let ${x}_{1},{x}_{2},\dots$ be a sequence in $X\text{.}$ Let $ℱ$ be the filter consisting of all subsets of $X$ which contain all but a finite number of points of $\left\{{x}_{1},{x}_{2},\dots \right\}\text{.}$ Show that $ℱ$ is a Cauchy filter if and only if ${x}_{1},{x}_{2},\dots$ is a Cauchy sequence.

Homework: Let $\left(X,𝒳\right)$ be a uniform space and let $ℱ$ be a filter on $X\text{.}$ Show that if $ℱ$ is convergent then $ℱ$ is Cauchy.

Homework: Let $\left(X,𝒳\right)$ be a uniform space and let ${x}_{1},{x}_{2},{x}_{3},\dots$ be a sequence in $X\text{.}$ Show that if ${x}_{1},{x}_{2},\dots$ is convergent then ${x}_{1},{x}_{2},\dots$ is a Cauchy sequence.

Homework: Give an example of a Cauchy sequence that does not converge.

Homework: Give an example of a Cauchy filter that does not have a limit point.

A complete space is a uniform space $\left(X,𝒳\right)$ such that if $ℱ$ is a Cauchy filter on $X$ then $ℱ$ has a limit point.

Homework: Let $\left(X,𝒳\right)$ be a complete space and let $Y\subseteq X\text{.}$ Show that if $Y$ is closed then $Y$ is complete.

Homework: Let $\left(X,𝒳\right)$ be a Hausdorff uniform space and let $Y\subseteq X\text{.}$ Show that if $Y$ is complete then $Y$ is closed.

Homework: Let $\left\{\left({X}_{i},{𝒳}_{i}\right) | i\in I\right\}$ be a collection of uniform spaces. Show that $\prod _{i\in I}{X}_{i}$ is complete if and only if the collection $\left\{\left({X}_{i},{𝒳}_{i}\right) | i\in I\right\}$ satisfies if $i\in I$ then ${X}_{i}$ is complete.

## Completions of uniform spaces

Let $\left(X,𝒳\right)$ be a uniform space.

A minimal Cauchy filter on $X$ is a Cauchy filter $ℱ$ on $X$ such that if $𝒢$ is a Cauchy filter on $X$ and $𝒢\subseteq ℱ$ then $𝒢=ℱ\text{.}$

The completion of $X$ is the set $Xˆ= {minimal Cauchy filters xˆ on X}$ with uniform structure $\stackrel{ˆ}{𝒳}$ generated by the sets $Vˆ= { (xˆ,yˆ)∈ Xˆ×Xˆ | there exists N∈xˆ∩ yˆ such that N×N ⊆V }$ for $V\in 𝒳$ such that if $\left(x,y\right)\in V$ then $\left(y,x\right)\in V$ and with the uniformly continuous map $i:X⟶Xˆgiven by i(x)=𝒩(x),$ where $𝒩\left(x\right)$ is the neighbourhood filter of $x\text{.}$

Homework: Show that the sets $\stackrel{ˆ}{V}$ for $V\in 𝒳$ such that if $\left(x,y\right)\in V$ then $\left(y,x\right)\in V$ generate a uniform structure on $\stackrel{ˆ}{X}\text{.}$

Homework: Show that $\stackrel{ˆ}{X}$ is Hausdorff.

Homework: Show that the uniform structure on $X$ is the inverse image of the uniform structure on $\stackrel{ˆ}{X}$ under $i:X\to \stackrel{ˆ}{X}\text{.}$

Homework: Show that $i:X\to \stackrel{ˆ}{X}$ is uniformly continuous.

Homework: Show that $\stackrel{ˆ}{X}$ is complete.

Homework: Show that $i\left(X\right)$ is dense in $\stackrel{ˆ}{X}\text{.}$

Homework: Show that if $Y$ is a complete Hausdorff uniform space and if $f:X\to Y$ is a uniformly continuous function then there exists a unique uniformly continuous function $g:\stackrel{ˆ}{X}\to Y$ such that $g\circ i=f$ $X ⟶f Y i↘ ↗g Xˆ$

## Notes and References

These are a typed copy of handwritten notes from the pdf 140721UniformSpacesscanned140721.pdf.