bigideas

In this instalment of our mini-series on topology we ask what makes a mapping between metric spaces continuous. The reason for asking this question is straightforward: topology is motivated by efforts to understand continuity in unfamiliar spaces, and to understand which properties can’t be changed by continuous mappings.

General Facts About Mappings

As with almost any mathematical object, it’s possible to define mappings (often called “functions”) between metric spaces. If A and B are metric spaces, a mapping from A to B assigns one point of B to each point in A. In this section we’ll look at some basic properties of all mappings.

For example, a geographical map is a metric space; it’s a plane whose points are the elements of the underlying set and we generally use the usual, Euclidean metric to measure distances. And the portion of the Earth’s surface that the map represents is also a metric space, again measured, usually, with the Euclidean metric (assume that the map isn’t of the whole Earth’s surface). There’s an obvious function that assigns to each point on the map the point on the Earth’s surface to which it corresponds. We call the map the “domain” of the function and the mapped portion of the Earth’s surface its “codomain”.

This function is called “one-to-one”, meaning each point in the codomain is mapped to by only one point in the domain (assuming the map is any good). It’s also “onto”, meaning every point in the codomain has a point in the domain mapped to it. It’s also “invertible”, meaning we can construct another function that sends points on the Earth’s surface to points on the map such that doing this new function after the original one leaves us back where we started. We’ll have more to say about invertible functions as we go along.

Now screw up the map into a big messy ball. We can define a function that assigns each point on the map to the point on the Earth’s surface that’s directly below it. This function is not one-to-one, because there will be several points on the map that are directly above or below one another, and that therefore get assigned to the same point on the floor. It isn’t onto either, because apart from the patch of floor under the map none of the other points in the codomain are mapped to.

Nor is it invertible, for two reasons. First, there are points in the codomain that have multiple points in the domain assigned to them; an inverse map would have to pick just one. Second, there are points in the codomain that have no points in the domain assigned to them at all, so what should the inverse map do with those?

These facts about one-to-one-ness and onto-ness are obviously related to invertibility. In fact, the relationship is simple and easy to understand:

A function has an inverse if and only if is it one-to-one and onto.

You may like to try convincing yourself of this or, if you feel so inclined, proving it rigorously. A bit of set theory is all you need.

It’s an interesting fact that in the case of the screwed-up map there will always be a point on the screwed-up map that’s directly above the point it corresponds to on the Earth’s surface. If you want to see what this is about, read this article about Brouwer’s Fixed Point Theorem.

Continuity: The General Idea

The main thing we, as topologists, care about when it comes to functions is whether they’re continuous or not. “Continuity” is hard to define; we have a sense that it might allow for stretching or bending, but not sticking or tearing, but how to formalise that isn’t obvious.

When a function maps a metric space to itself it’s a bit easier to understand; then the function is a “transformation” of the metric space. When we crumpled up the map, was that a continuous transformation or a discontinuous one? What about if we’d torn a piece off and stuck it back upside-down?

We’re inclined to call the transformations in this video “continuous”:

although the fact that the transformations is animated is a bit misleading; a function takes you straight from the first frame to the final one, without being able to see what happens in between.

The usual intuitive picture of a continuous map is one in which “points that start off close together end up close together”, but that on its own isn’t too helpful. Another way of putting it is that “open sets don’t get too mangled”, but that doesn’t help much more. Perhaps another way is to say that we can imagine an animation of a continuous map that looks “smooth” like the animation of the swans above, that doesn’t have any tearing in it. A smooth function from a balloon to itself is a transformation you can do without bursting the balloon.

Hmmm… we clearly need to get something formal in place here.

Continuity: The Formal Definition

The formalism mathematicians have come up with is the following. We’ll explain what it means in a second:

A function f that maps A to B is continuous if and only if for every open set U in B, f^-1(U) is an open set in A.

First, there’s a bit of odd notation here, which is this:

f^-1(U)

All this means is “the subset of A in which every element is mapped to an element of U by f”.

Take the example of the screwed-up map and the area of the floor directly under it. now pick an open subset of that area — say a circular spot as big as a ten pence piece, not including the edge (remember what we said last time about boundary points?). Call this U. Now f^-1(U) consists of all the points on the map directly above U. In fact if you were somehow able to colour these in and then flatten the map out again, there would most likely be several coloured patches that, taken together, correspond to the set f^-1(U).

The definition says that the function from the “crumpled up” map to the floor is continuous if and only if these coloured patches together make up an open set, that is, if every point in them has an open disk centred on it that’s wholly contained in the coloured patch.

Why did mathematicians settle on this definition and not some other one? The simple answer is that is turns out to capture most of our intuition about what it means for a mapping to be continuous. I hope that somewhere in the course of this mini-series, after seeing lots of examples and non-examples of continuous mappings, that you’ll come to agree.

That’s all we need from metric spaces: open sets and continuous maps. In the next instalment we’ll put them aside and begin our trip into the more abstract world of topology proper.

This post is a part of a Topology Mini-Series