In my introduction, I said that topology is fundamentally built on the notion of closeness. Someone very quickly responded on Twitter, because they thought that was wrong. It wasn’t wrong, but it’s easy to see where the confusion came from. Like so much math, Topology is built on a very precise logical and set-theoretic formalism. Mathematicians build those formalisms not because they’re annoying people who want to be mysterious and incomprehensible, but because the precision of those formalisms is critically important.
When you hear a statement like “point A is close to point B in a space S”, you have an intuitive idea of what the word “close” means. But when you try to expand that to math, it could potentially mean several different things. The easiest meaning would be: the distance between A and B is small.
Mathematicians have used that definition for a lot of interesting work. It’s got one limitation though: For it to work, you need to be able to define “distance” in the space. How do you do that? In conventional Euclidean space, we have an easy definition. Describe the position of the two points using Cartesian coordinates: A=(x1, y1), B = (x2, y2). The distance between A and B is:
But we’re moving towards the world of topology. We can’t count on our spaces to be Euclidean. In fact, the whole point of topology is, in some sense, to figure out what happens when you have different spatial structures – that is, structures other than the familiar Euclidean one! We need to be able to talk about distances in some more general way. To do that, we’ll create a new kind of space – a space with an associated distance metric. This new space is called a metric space.
A distance metric is conceptually simple. It’s just a special kind of function, from pairs of points in a space to a real number. To be a distance metric, it needs a couple of properties. Suppose that the set of points in the space is . Then a function is a distance metric if it satisfies the following requirements:
- Triangle Inequality:
A metric space is just the pair of a set , and a metric function over the set. For example:
- A cartesian plane is a metric space whose metric function is the euclidean distance: .
- A checkerboard is a metric space with the number of kings moves as the metric function.
- The Manhattan street grid is a metric space where the distance function between two intersections is the sum of the number of horizontal blocks and the number of vertical blocks between them.
All of this is the mathematical work necessary to take one intuitive notion of closeness – the idea of “two points are close if there’s a small distance between them” and turn it into something formal, general, and unambiguous. But we still haven’t gotten to what closeness means in topology! It’s not based on any idea of distance. There are many topological spaces which aren’t metric spaces – that is, there’s no way to define a metric function!
Fortunately, metric spaces give us a good starting point. In topological spaces, closeness is defined in terms of neighborhoods and open balls.
Take a metric space, , and a point . An open ball B(p, r) (that is, a ball of radius around the point ) is the set of points .
Given a large enough set of points, you can create an infinite series of concentric open spheres: , and so on. Once you’ve got that series of ever-smaller and ever-larger open balls around a point , you’ve got another notion of closeness. is closer to than is if is in a smaller open ball around .
This is the heart of topology. You can define something like an open-ball on a set without a metric. As long as you can create a consistent sequence of open balls, where each larger ball is a strict superset of all of the smaller ones, you can define closeness without any notion of measurable distance!
In the next post, we’ll use this notion of a distance-free sense of closeness to define what a topology actually is.