bigideas

Now we have the notion of a homeomorphism under our belts, and the idea that homeomorphic spaces are topologically equivalent, we’re in a position to decide whether certain properties of Euclidean spaces are or aren’t topological.

As we do that, it will be useful to have some more examples of topological spaces. We’ll use the next few instalments to demonstrate some different techniques for constructing new topological spaces from the ones we already have. We’ll particularly focus on the real line R, since that’s the main source of the kinds of spaces we’ll look at later.

The Subspace Topology

Let (S, T) be a topological space and X be a subset of S. Now take each element of T, find its intersection with X and put it into a new set, say Y. We get (X, Y), a new topological space, and Y is called the subspace topology on X.

A simple example is a “closed interval” of the real line R, meaning two numbers, say 0 and 1, and all the numbers in between them. We’ll write it in the standard way, that is, as [0, 1]. The open sets of this space are all the open sets of R that intersect with it.

It includes all the open balls that are subsets of [0, 1], and all their unions and (finite) intersections, plus all the parts of open balls that overlap [0, 1]. Those are not open balls in R itself. Remember, the open balls are intervals (a, b) that don’t include their end-points. But look at the intersection of, say, (0.5, 1.5) with [0, 1] — it’s the “half-open” interval (0.5, 1] that contains one of its end-points — 1 — but doesn’t contain the other. This is an open set in the subspace topology on [0, 1], but it isn’t an open set in R.

Size

Let’s consider some maps from [0, 1] to other closed intervals of R. For a start, let f map [0, 1] to [0, 2] by the following rule:

f(x) = 2x

So f(0)=0, f(1/2)=1, f(1)=2 and so on. It’s not hard to see intuitively that any open set in [0, 2] is going to be the image of an open set in [0, 1] that’s just a bit smaller. If you’re so inclined, you might want to think about how a rigorous proof of this might go; think about the open and half-open intervals in [0, 2] and what sets map to them from [0, 1], then worry about their unions and intersections later.

So f is continuous; but is it a homeomorphism? It is, because we have an inverse map from [0, 2] to [0, 1] with the rule

f^-1(x) = x/2

which very obviously undoes what f did. Relying on the same sort of intuition, this inverse is continuous, and so f is a homeomorphism. Hence [0, 1] and [0, 2] are topologically speaking exactly the same.

Since [0, 1] and [0, 2] are different sizes, but they’re topologically equivalent, we can conclude that size is not a topological property. You probably already knew, or guessed, as much from the idea of topology as “rubber sheet” geometry.

Boundedness

This time, take the open interval (0, 1) with the subset topology — in this case, there are no half-open intervals and all the open sets of (0, 1) are also open sets of R (why?). Now consider the map

f(x) = 1/x

The image of this map is the interval (1, infinity); with the subset topology on that map as well, f(x) is continuous and is its own inverse (that is, f(f(x)) = x).

We say that (0, 1) is a bounded space because there are points on either side of it that are greater and less than any points within it; indeed, it’s bounded by the points 0 on the left and 1 on the right. The space (1, infinity), on the other hand, is not bounded, because there’s no point on the right that’s greater than any point in it.

(Important: infinity isn’t a number, or at least it isn’t a part of the real line R. So infinity is not an upper bound for (1, infinity) in R. Technically speaking, there are many cases like this in which we can only talk about a space having bounds in the context of a larger space.)

Since f is a homeomorphism between a bounded space and an unbounded one, we can safely say that boundedness is not a topological property, perhaps contrary to what your intuition would tell you.

This post is a part of a Topology Mini-Series