bigideas

We’ve spent most of our time so far in metric spaces, which might seem odd for a series about topology. In this instalment we show how a metric space can be turned into a topological space, and how the idea generalises far beyond metric spaces.

The Basic Definition

You’ll notice that the definition of a continuous mapping was done purely in terms of open sets, and so it looks as if open sets give us everything we need to talk about continuity. We’d like our open sets to have certain properties, though.

For a start, we defined an open set in a metric space as a union of open balls. It follows that the union of two open sets is also an open set. We also want the intersection of open sets to be open again, but there’s a detail here that won’t be immediately clear.

We actually want the intersection of finitely many open sets to be an open set, but we allow the union of finitely or infinitely many open sets to be an open set. If you’ve done a course on analysis you’ll realise that allowing infinite intersections in Euclidean spaces would make single points open, and that’s not what we want at all. If you haven’t then it’s probably better to just accept this for now; explaining it would take us too far away from our main topic.

Indeed, these are just about all the requirements for a topology. Let’s state them formally:

Given a set S, a topology is a collection T of subsets of S such that:

• S and the empty set are in T
• All finite intersections of elements of T are again in T
• All finite or infinite unions of elements of T are again in T

the pair (S, T) is called a “topological space”.

This doesn’t sound like much, does it? But it turns out to be an extremely rich structure. Before we see that, let’s look at some simple examples.

Some Simple Examples of Topological Spaces

The Indiscrete Topology

Let S be any set. Then the set {S, {}}, containing just S and the empty set, is a topology on S. This is the “coarsest” topology on S — that is, the one with the fewest open sets — and as you might imagine it’s not very useful.

It’s usually known as the “indiscrete topology”, because you can’t separate any points or sets using open sets; the only open set with anything in it is all of S (we’ll talk a lot more about separation of points by sets shortly).

The Discrete Topology

Again, let S be any set. Then 2^S, the power set of S, is also a topology on S. Since it contains every subset of S, this topology has the largest possible number of open sets and is called the “finest” topology on S, known as the “discrete” topology.

The name comes from the fact that every “singleton” set {x} is open; as we mentioned above, this isn’t usually the case for topological spaces.

A Little Example

Let S be the set {a, b, c}, and T the set {{a, b, c}, {b, c}, {c}, {}}. Then (S, T) is a topological space that’s neither the coarsest nor the finest.

A nice visual way to represent topologies on very small sets is to draw circles around the elements to indicate the open sets, like this which illustrates the present example:

This becomes unwieldy very quickly, but we’ll occasionally use it because it’s certainly nice and visual.

The Alexandrov Topology

Let’s extend the previous example. Let S be any set that’s totally ordered, by which we mean that for every pair of elements x and y, either x ≤ y or y ≤ x according to some ordering that we designate by “≤”. The elements of S might not be numbers, but we may as well assume they are.

Now, let the open sets be all the sets of the form

{x : x < p}

for each p in S. In words, each open set contains some element of S and every other element that’s less than it. You can check for yourself that these sets, taken all together, form a topology on S. It’s called the Alexandrov topology and has important applications in logic and computer science.

By ordering the set {a, b, c} with the relation ≤ such that b≤a, c≤b and c≤a, we see that the previous example is just the Alexandrov topology in disguise. By choosing another ordering, you can create another Alexandrov topology; for instance, {{a, b, c}, {a, b}, {a}, {}}. How many different Alexandrov topologies are there on a three-element set? What about for an n-element set?

Metric Spaces

It should be fairly obvious that the definition makes all metric spaces topological spaces. The point was to create a generalisation, so you’d hope that was true. We just let the “open sets” of the topological space be the set of all “open sets” (in the metric space sense) of the metric space. The underlying set of the metric space, meanwhile, becomes the underlying set of the topological space. So a metric space (S, d) becomes a topological space (S, T), where T is all the (metric space) open sets given by the metric d. Notice that singletons in a metric space aren’t open, and so the topology isn’t discrete. This is good, because certain things wouldn’t work well if it were.

Notice what we’ve done here. We’ve started with a metric space, (S, d), composed of an underlying set (such as R²) and a distance measure d, the metric. We’ve taken the open sets that we used the metric to define and used them to define a topological space (S, T). In doing so we’ve thrown away the metric itself. This is quite deliberate, as we’ll explain in the next instalment.

 

This post is a part of a Topology Mini-Series