I'd like to learn a bit about uniform spaces, why are they useful, how do they arise, what do they generalize, etc., without getting away from the context of general topology. I have to prepare an 1h30min talk on the subject, for an audience formed in standard general topology (i.e. Munkres), not so much in abstract algebra (so I'd like not to use topological groups).

The references I have are Kelley or Willard texts on Topology, Isbell's "Uniform Spaces" and James's "Topological and Uniform Spaces".

I discarded the last one because of its heavy use of filters from the beginning. I don't know about any other good references. What I'd like to see is the subject treated as Munkres does in his book: he gives good motivations, pictures, and is gentle to the reader.

• I think that Kelley is a pretty good first introduction to uniform spaces -- certainly there's more than 90 minutes' worth of material there. As I recall, he gives more attention to the gauge definition of uniform space than most other references do, but that also makes his treatment more valuable. As I said in another MO answer, I think uniform spaces are hard to learn at the beginning due to the phenomenon of cryptomorphism -- i.e., there are at least three fully equivalent, but not obviously so, ways to formalize the concept. – Pete L. Clark Aug 28 '10 at 21:21
• I should add that I am not as fully conversant with this material as I would like to be, so I am looking forward to a good answer to this question. Someone is surely going to say to look in Bourbaki, and I think that's not a wrong answer, but I would honestly prefer to read a digested revision of the theory as presented there. – Pete L. Clark Aug 28 '10 at 21:23
• You need to give good examples or people may not understand what you're talking about or why anyone should care. Uniform spaces serve to combine metric spaces and topological groups in one setting, namely an abstract context in which one can talk about uniform continuity. Even if they haven't had abstract algebra, I think you have to make an effort at giving some concrete examples from topological groups (say, Z with its p-adic topology, defined algebraically using congruence conditions without saying "p-adic metric", and the function f(a) = a^5 on Z is p-adically unif. continuous). – KConrad Aug 28 '10 at 21:34
• -1. "I discarded the last one because of is heavy use of filters from the beginning." Um. I hope you know that the entourage definition of a uniformity is defined as a FILTER on $X\times X$ that satisfies the properties of entourages. Furthermore, the covering definition of a uniformity is defined as a FILTER on the preordering of all coverings of a uniform space. Filters are ubiquitous in general topology and in the theory of uniform spaces. – Joseph Van Name Oct 13 '15 at 2:34
• Joseph, that doesn't seem like a very good reason for a downvote. The question makes clear that this is for a 90 minute talk. If I sit in on a 90 minute talk where I have to learn one abstract concept in order to understand another abstract concept, I don't end up learning two abstract concepts. I instead learn zero abstract concepts. – arsmath Aug 22 '17 at 17:45

I would motivate them as follows: if topological spaces were invented to give a general meaning to "continuous function", then uniform spaces were invented to give a general meaning to "uniformly continuous function". It is clear what "uniformly continuous" should mean for metric spaces and topological groups, but how should the general notion be formalized?

Once this is formalized, one can define the notion of Cauchy net in a uniform space (which is something you cannot do for general topological spaces). This leads to the notion of completeness of course (every Cauchy net converges to at least one point), although the theory is much cleaner for complete Hausdorff uniform spaces, where you have convergence to at most one point as well.

To illustrate this: the Cauchy completion of a uniform space $X$ can be defined in the usual way via equivalence classes of Cauchy nets. It is a complete Hausdorff uniform space $\bar{X}$ together with a map $i: X \to \bar{X}$ which satisfies a universal property: given a complete Hausdorff uniform space $Y$ and a uniformly continuous function $f: X \to Y$, there is a unique uniformly continuous map $\bar{f}: \bar{X} \to Y$ such that $\bar{f} \circ i = f$. (If you omit "Hausdorff" or "uniformly", you lose the universal property, which is arguably the point of the completion.)

The nLab has an article on uniform spaces with some material not included in the Wikipedia article.

• nLab link: ncatlab.org/nlab/show/uniform+space – François G. Dorais Aug 29 '10 at 15:17
• The motivation you give is the explanation of how I bumped into uniform spaces. I was studying general topology, and uniformly continuous functions arose here and there (especially when studying complete metric spaces). I have a little formation on category theory (and I'm quite inclined to it), so I tend to try and organize information from this viewpoint. I asked myself this question: if continuous maps are the morphisms in Top, and weak contractions are the morphisms in Met, for which objects are uniformly continuous maps morphisms? If I have not misunderstood, it would be uniform spaces. – Bruno Stonek Aug 29 '10 at 15:31

For a general audience it can be interesting to know that uniform spaces are just one of two opposite generalizations of metric spaces. Measuring distances with the help of metric, we can be interested in points lying on small distance (and exactly this aspect interests specialists in uniform spaces) but also is points lying on large distance (this is the subject of coarse geometry). So, uniform and coarse spaces are two opposite generalizations of metric spaces, corresponding to two ends of the real half-line $(0,\infty)$. Both of them allow us to compare distances between points without using real numbers.

Uniform spaces and coarse spaces have dual definitions. Both of them are pairs $(X,\mathcal U$) consisting of a set $X$ and a family $\mathcal U$ of subsets $U\subset X\times X$ containing the diagonal but uniform spaces obey the axioms:

(U1) for any $U\subset V\subset X\times X\quad U\in\mathcal U$ implies $V\in\mathcal U$;
(U2) each $U\in\mathcal U$ the entourage $-U:=\{(y,x):(x,y)\in U\}$ belongs to $\mathcal U$;
(U3) each $U\in\mathcal U$ contains $V+V:=\{(x,z):\exists y\in X\;(x,y),(y,z)\in V\}$ for some $V\in\mathcal U$.

Whereas coarse spaces obey the dual axioms:

(C1) for any $U\subset V\subset X\times X\quad V\in\mathcal U$ implies $U\in\mathcal U$;
(C2) each $U\in\mathcal U$ the entourage $-U:=\{(y,x):(x,y)\in U\}$ belongs to $\mathcal U$;
(C3) for each $U\in\mathcal U$ the set $U+U$ is contained in some $V\in\mathcal U$.

There is also a duality in morphisms. For the category of uniform spaces morphisms are uniformly continuous maps, i.e., functions $f:X\to Y$ between uniform spaces $(X,\mathcal U_X)$ and $(Y,\mathcal U_Y)$ such that for any $U\in\mathcal U_Y$ there exists $V\in\mathcal U_X$ such that $\{(f(x),f(y)):(x,y)\in V\}\subset U$.

For the category of coarse spaces morphisms are coarse maps, i.e., functions $f:X\to Y$ between coarse spaces $(X,\mathcal U_X)$ and $(Y,\mathcal U_Y)$ such that for any $V\in\mathcal U_X$ there exists $U\in\mathcal U_Y$ such that $\{(f(x),f(y)):(x,y)\in V\}\subset U$.

In fact, the structure of a uniform (coarse) space is the most general structure which allows us to speak about uniformly continuous (resp. coarse) maps whose $\varepsilon$-$\delta$-definitions were known in analysis long before the appearence of uniform (or coarse) spaces. And this was the reason for introducing uniform or coarse spaces.

Concerning the references, for me the basic reference in uniform spaces is Chapter 8 of Engelking's book "General topology" and for coarse spaces the book "Lectures on coarse geometry" of J.Roe.

For a bit deeper questions related to uniform or coarse spaces, which can be interesting for general audience, I would recommend the problem of dimension. Why does the space $\mathbb R^n$ have dimension $n$ and in which sense? In fact, there are many deep answers to this question in many various categories having $\mathbb R^n$ as an object (in particular, in the category of linear spaces, rings, metric spaces, topological spaces, etc).

The categories of uniform and coarse spaces also have their own definitions of dimension, which are dual in some sense and yield the same dimension $n$ for the space $\mathbb R^n$.

The uniform dimension $dim(X,\mathcal U)$ of a uniform space $(X,\mathcal U)$ is the smallest number $n$ such that for any $U\in\mathcal U$ there exist $V\in\mathcal U$ and a cover $\mathcal C$ of $X$ by sets $C\subset X$ of diameter $<U$ (which means that $C\times C\subset U$) such that $\mathcal C$ can be written as the union $\mathcal C=\bigcup_{i=0}^n\mathcal C_i$ of $(n+1)$ subfamilies $\mathcal C_i$, which are $V$-separated in the sense that $(C\times D)\cap V=\emptyset$ for any distinct sets $C,D\in\mathcal C_i$.

The coarse (or else Gromov's asymptotic) dimension $Dim(X,\mathcal U)$ of a coarse space $(X,\mathcal U)$ is the smallest number $n$ such that for any $V\in\mathcal U$ there exist $U\in\mathcal U$ and a cover $\mathcal C$ of $X$ by sets $C\subset X$ of diameter $<U$ such that $\mathcal C$ can be written as the union $\mathcal C=\bigcup_{i=0}^n\mathcal C_i$ of $(n+1)$ subfamilies $\mathcal C_i$, which are $V$-separated.

These two definition have only one change in quantifiers but "look" in different directions -- uniform dimension takes small $U$, but the coarse direction takes large $V$!

The standard wall tiled with (small or large) bricks shows that the plane has both uniform and coarse dimension equal to 2. A non-trivial result of uniform and coarse dimension theories says that the uniform and coarse dimension of $\mathbb R^n$ is equal to $n$. By the way, the coarse dimension of $\mathbb Z^n$ also is $n$ whereas the uniform dimension of $\mathbb Z^n$ is zero. On the other hand, the uniform dimension of the cube $[0,1]^n$ is $n$ but the coarse dimension of $[0,1]^n$ is zero.

• Thank you for posting this! This analogous presentation of uniform and coarse spaces is particularly enlightening. (But now I am tempted to ask whether there exists something interesting which is to coarse spaces what topological spaces are to uniform spaces.) – Gro-Tsen Aug 22 '17 at 18:30
• @Gro-Tsen Yes, there exists -- this is a bornology, i.e. an ideal of subsets, which can be thought as bounded sets, but this is very poor structure, which can only distinguish bounded sets from non-bounded. It can also feel the cofinality of the ideal of bounded sets. In any case it is quite far from the topology and without such exciting symmetry as in case of uniform versus coarse spaces. – Taras Banakh Aug 22 '17 at 19:00

To sell uniform spaces in a single presentation requires above all some convincing motivation. Why develop all this machinery as a generalization of metric spaces? What nontrivial examples require such an artificial-looking development?

The Wikipedia entry here is accurate as far as it goes, though as usual the references and examples are inadequate. As that article points out, uniform spaces were introduced by Weil in 1937 (applied in his 1940 monograph L'integration dans les groupes topologiques et ses applications). Bourbaki gave a reasonable but rather formal treatment of the foundations in early chapters of their book on general topology.

Without topological groups as examples, it's tough to offer enough motivation. For me the reason to look at uniformities was the need to understand the approach of Serre and others (Bass, Milnor, Matsumoto, Prasad, Raghunathan) to the Congruence Subgroup Problem for algebraic groups over number fields such as SL$(n,\mathbb{Q})$. Here one has two natural subgroup topologies, with fundamental systems of neighborhoods of the identity given by all arithmetic subgroups or just by congruence subgroups. Profinite completion by itself isn't enough to formulate the problem precisely, so completion relative to a uniform structure comes into play. There is an elementary introduction to these ideas in Section 16 of my old Springer Lecture Notes 789 Arithmetic Groups.

• I think this is exactly right. One should have studied topological groups first. And of course metric spaces. One sees a "completion" in metric spaces. Then in topological groups. But not in general topology. So what is going on? NOW one is motivated to study uniform spaces. – Gerald Edgar Aug 29 '10 at 13:01
• I understand topological groups may be good prior knowledge to motivate uniformities. My talk is in a seminar on general topology. The audience knows about completeness in metric spaces, and will know about metrization. Maybe I could try to get to talk after someone who introduces topological groups. – Bruno Stonek Aug 29 '10 at 13:45

How do they arise &c: metrization theorems are certainly a natural source. Under this point of view, the "uniformization" of a topology may be seen as a first main step towards metrization. Furthermore, topological groups and topological vector spaces are very natural examples of uniform spaces that are not necessarily metrizable. Actually, they could provide a nice source of examples for your seminar too; some theorems or constructions about uniform spaces take a particularly simple form in the case of TG and TVS. You may e.g. sketch the construction of a metric for a first-countable Hausdorff TVS (if, in such a topological vector space $X$, the family $\{U_n\}_n$ is a base of symmetric neighborhoods of the origin such that

$U_{n+1}+U_{n+1}\subset U_n$, define for any $x\in X$ the quantity $q(x)$ to be the infimum of $\sum_i 2^{-k_i}$ taken over all finite sequences $(k_1,k_2,\dots, k_r)$ such that $x\in U_{k_1}+U_{k_2}+\dots+U_{k_r},$ and prove that $d(x,y):=q(x-y)$ metrizes $X$).

Why not try Weil's original paper: it's reference 12 in this paper.

About motivation. (Late answer, sorry) I don't know what you have chosen and how it went, but had I have found your question in time, I would have recommended pseudometrics. Your audience knows metric spaces ? they must feel very well pointwise convergence then (say for real-valued functions on a set $X$). But, if the base set $X$ is not denumerable it cannot be explained by a single metrics but by very simple pseudometrics (defined through the coordinate functions).

About place to learn uniform spaces. I would recommend Bourbaki, General Topology, because they give as well the equivalence with pseudometrics, the categorical (without saying) view of completion and the motivation by topological groups.