bigideas

We continue our introduction to topology with a look at some of the properties of the metric spaces we defined in the previous instalment. We’re particularly interested in defining a special kind of subset of a metric space that we’ll need constantly later. Remember, in all that follows we’re working with metric spaces; topological spaces are more general, and will require more abstract formulations to get similar results.

Open Balls

The key to capturing the essence of continuity turns out to the the idea of an open ball. Open balls are defined in terms of metrics. An open ball has two properties — its radius (say, r) and a central point (say, x). It’s simply the set of all points p such that

d(x, p) < r

Notice that the distance must be less than r — any points that are exactly r away from x aren’t included.

Examples

In Euclidean space, open balls really are (sort of) ball-shaped. Let’s take the plane R². Imagine a goat lives on the plane, and he’s tethered to a post by a rope that’s a bit uncomfortable for him when it’s fully extended. How much of the grass can he comfortably eat? He can eat anything in the circle around the post, going out almost but not quite as far (in all directions) as the length of the rope.

This, without the post and the goat and the grass, is an open ball in R². Of course, x is the place where the pole goes, and mathematically speaking the goat can eat the grass there, too.

It doesn’t take much thinking to see that open balls in the number line R are “intervals” — sections of the line with x in the centre, and without their “end-points”. The open ball in R of radius 1 about 3, for instance, is the interval (2, 4), which consists of all the numbers between 2 and 4, but not 2 or 4 themselves. Open balls in R³ are actually ball-shaped; specifically, they’re the insides of spheres.

Note that an open ball is defined with reference to a point and also a metric, so if the metric changes the open balls might change too. Here’s an exercise for you: in the taxicab metric, when shape are the open balls?

General Open Sets

We can generalise the idea of open balls a bit by defining an “open set” in a metric space to be any set that can be considered as a union of open balls. We won’t try to prove it, but I hope you can imagine that in Euclidean space you can make all kinds of shapes by packing smaller and smaller open balls together. You can probably see that, in “normal” cases, the whole space is going to be an open set. When we generalise this into topological terms, we’ll specifically require that.

Intuitively, one very important thing about an open ball was the fact that it didn’t include its boundary (because the little goat that makes the open ball doesn’t like to stretch the rope completely tight). Open sets in Euclidean space are like this too; they can have strange shapes, but they never include their boundaries.

A more rigorous way to express this is by the following definition:

In Euclidean n-space, a set S is open if every point in S has an open ball centred on it that’s entirely contained in S.

If the point is very close to the edge of S then the open ball has to be very small, but you can always find one.

If S includes any boundary points, on the other hand, no open ball around them is ever fully contained in S because part of it, by definition, will fall on the other side of the boundary.

Recall that the space underlying the taxicab metric is a set of discrete corners, like (43, 4) being the corner of 43rd street and 4th avenue. The unions of open balls in this space don’t have the property just described for Euclidean spaces. This is a direct consequence of the fact that this space is composed of discrete points (corners) rather than a continuum. We’ll talk more about that later.

As we’ll see in the next instalment, open sets are extremely useful in helping us to define a continuous mapping between metric spaces — and it’s that definition that we’ll want ot generalise when we come to define a topological space. Indeed, a topological space is defined by specifying its open sets, although in that context an “open set” might look very different to the ones described here.

This post is a part of a Topology Mini-Series