Uniform spaces

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

Last update: 23 July 2014

Uniform spaces

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

(a)	if $V\subseteq X\times X$ and $B\in 𝒳$ and $B\subseteq V$ then $V\in 𝒳\text{,}$
(b)	if $\ell \in {\mathbb{Z}}_{>0}$ and $V_1,V_2,\dots ,V_{\ell }\in 𝒳$ then $$V_1\cap V_2\cap \cdots \cap V_{\ell }\in 𝒳\text{,}$$
(c)	if $V\in 𝒳$ then $\left\{(x,x)\hspace{0.17em}\right|\hspace{0.17em}x\in X\}\subseteq V\text{,}$
(d)	if $V\in 𝒳$ then $\left\{(y,x)\hspace{0.17em}\right|\hspace{0.17em}(x,y)\in V\}\in 𝒳,$
(e)	if $V\in 𝒳$ then there exists $M\in 𝒳$ such that $M{\times }_{X}M\subseteq V$ where $$M{\times }_{X}M=\left\{(x,y)\hspace{0.17em}\right|\hspace{0.17em}\text{there exists}\hspace{0.17em}z\in X\hspace{0.17em}\text{such that}\hspace{0.17em}(x,z)\in M\hspace{0.17em}\text{and}\hspace{0.17em}(z,y)\in M\}\text{.}$$

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

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

Let $(X,𝒳)$ be a uniform space. The uniform space topology on $X$ is the topology on $X$ such that if $x\in X$ then $$𝒩\left(x\right)=\{B_V\left(x\right)=\{y\in X\hspace{0.17em}|\hspace{0.17em}(x,y)\in V\}\hspace{0.17em}|\hspace{0.17em}V\in 𝒳\}$$ is the neighbourhood filter of $x\text{.}$

Homework: Let $(X,𝒳)$ and $(Y,𝒴)$ 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 $(X,𝒳)$ 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 $N\times N\subseteq V\text{.}$

A Cauchy sequence is a sequence $x_1,x_2,\dots$ in $X$ such that if $V\in 𝒳$ then there exists $n_0\in {\mathbb{Z}}_{>0}$ such that if $m,n\in {\mathbb{Z}}_{>0}$ and $m>n_0$ and $n>n_0$ then $(x_m,x_n)\in V\text{.}$

Homework: Let $(X,𝒳)$ 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 $\{x_1,x_2,\dots \}\text{.}$ Show that $ℱ$ is a Cauchy filter if and only if $x_1,x_2,\dots$ is a Cauchy sequence.

Homework: Let $(X,𝒳)$ 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 $(X,𝒳)$ 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 $(X,𝒳)$ such that if $ℱ$ is a Cauchy filter on $X$ then $ℱ$ has a limit point.

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

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

Homework: Let $\left\{(X_i,{𝒳}_i)\hspace{0.17em}\right|\hspace{0.17em}i\in I\}$ be a collection of uniform spaces. Show that $\prod _{i\in I}{X}_i$ is complete if and only if the collection $\left\{(X_i,{𝒳}_i)\hspace{0.17em}\right|\hspace{0.17em}i\in I\}$ satisfies if $i\in I$ then $X_i$ is complete.

Completions of uniform spaces

Let $(X,𝒳)$ 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 $$\hat{X}=\left\{\text{minimal Cauchy filters}\hspace{0.17em}\hat{x}\hspace{0.17em}\text{on}\hspace{0.17em}X\right\}$$ with uniform structure $\widehat{𝒳}$ generated by the sets $$\hat{V}=\{(\hat{x},\hat{y})\in \hat{X}\times \hat{X}\hspace{0.17em}|\hspace{0.17em}\text{there exists}\hspace{0.17em}N\in \hat{x}\cap \hat{y}\hspace{0.17em}\text{such that}\hspace{0.17em}N\times N\subseteq V\}$$ for $V\in 𝒳$ such that if $(x,y)\in V$ then $(y,x)\in V$ and with the uniformly continuous map $$i:X⟶\hat{X}\text{given by}i\left(x\right)=𝒩\left(x\right),$$ where $𝒩\left(x\right)$ is the neighbourhood filter of $x\text{.}$

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

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

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

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

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

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

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 $g:\hat{X}\to Y$ such that $g\circ i=f$ $$\begin{array}{ccccc}X& \multicolumn{3}{c}{\stackrel{f}{⟶}}& Y\\ & i\searrow & & \nearrow g& \\ & & \hat{X}& & \end{array}$$

Notes and References

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

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