Now let’s add a little more structure to our topological spaces. We can use a topology on a set to talk about which points are “close” to a subset. Now we want to make a finer comparison by being able to say “the point is closer to the subset than is to .” We’ll do this with a technique similar to neighborhoods. But there we just defined a collection of neighborhoods for each point. Here we will define the neighborhoods of all of our points “uniformly” over the whole space.
To this end, we will equip our set with a family of subsets of called the “uniform structure” on our space, and the elements will be “entourages”. We will write for the set of so that , and we want these sets to form a neighborhood filter for as varies over . Here we go:
- Every entourage contains the diagonal .
- If is an entourage and , then is an entourage.
- If and are entourages, then is an entourage.
- If is an entourage then there is another entourage so that and imply .
- If is an entourage then its reflection is also an entourage.
The first of these axioms says that , as we’d hope for a neighborhood. The next two ensure that the collection of all the forms a neighborhood filter for , but it does so “uniformly” for all the at once. This means that we can compare neighborhoods of two different points because each of them comes from an entourage, and we can compare the entourages. The fourth axiom is like the one I omitted from my discussion of neighborhoods; every collection of entourages gives rise to a topology, but topologies can only give back uniform structures satisfying this requirement. Finally, the last axiom gives the very reasonable condition that if , then . That is, if one point is in a neighborhood of another, then the other point should be in a neighborhood of the first. Sometimes this requirement is omitted to get a “quasi-uniform space”.
Now that we can compare closeness at different points, we can significantly enrich our concept of nets. Before now we talked about a net converging to a point in the sense that the points eventually got close to . But now we can talk about whether the points of the net are getting closer to each other. That is, for every entourage there is a so that for all and the pair is in . In this case we say that the net is “Cauchy”.
Now, if the full generality of nets still unnerves you, you can restrict to sequences. Then the condition is that there is some number so that for any two numbers and bigger than we have . This gives us the notion of a Cauchy sequence, which some of you may already have heard of.
We can also enrich our notion of continuity. Before we said that a function from a topological space defined by a neighborhood system to another one is continuous at a point if for each neighborhood contained the image of some neighborhood , and we said that was continuous if it was continuous at every point of .
Now our uniform structures allow us to talk about neighborhoods of all points of a space together, so we can adapt our definition to work uniformly. We say that a function from a uniform space to another one is uniformly continuous if for each entourage there is some entourage that gets sent into . More precisely, for every pair the pair is in .
In particular, any neighborhood of a point is of the form for some entourage . Then uniform continuity gives us an entourage , and thus a neighborhood which is sent into . Thus uniform continuity implies continuity, but not necessarily the other way around. It is possible that a function is continuous, but that the only ways of picking neighborhoods to satisfy the definition do not come from entourages.
These two extended definitions play well with each other too. Let’s consider a uniformly continuous function and a Cauchy net in . Then I assert that the image of this net is again Cauchy. Indeed, for every entourage we want a so that and imply that the pair is in . But uniform continuity gives us an entourage that gets sent into , and the Cauchy property of our net gives us a so that for all and above . Then and we’re done.
It wouldn’t surprise me if one could turn this around like we did for neighborhoods. Given a map which is not uniformly continuous use the uniform structure as a directed set and construct a net on it which is Cauchy in , but whose image is not Cauchy in . Then one could define uniform continuity as preservation of Cauchy nets and derive the other definition from it. However I’ve been looking at this conjecture for about an hour now and don’t quite see how to prove it. So for now I’ll just leave it, but if anyone else knows the right construction offhand I’d be glad to hear it.