bigideas

We ended the previous instalment with the idea that a topological space is a metric space with the metric “thrown away” (not quite, but in a manner of speaking) and we suggested that this was no coincidence. In this instalment we’ll get to see some of the fruits of our labour by defining one of the most important concepts in topology.

Homeomorphisms

You may have heard of topology described as “rubber sheet geometry“, a geometry in which we can stretch and distort things as much as we like and they remain “the same”. Well, this is the result of discarding the metric, and this “sameness” is expressed in terms of a special map called a “homeomorphism“.

In a metric space, a continuous map that, say, stretches a circle into a square changes something important, because it changes the distances between points. But in a topological space, all the open sets just get deformed a bit; they all contain the same points at the end as they did at the start, so they’re identical (that’s what the set-theoretic axiom of extensionality is for; it asserts that two sets with the same members are identical).

This means that continuous maps that leave the open sets intact don’t change the topological space at all. Remember, the topological space just is the open sets; nothing else. These maps are homeomorphisms. Intuitively, they stretch the rubber sheet, or squash the modelling clay, of a topological space without doing any damage to it, and they can always be reversed if desired.

Remember, a continuous map f between two metric spaces A and B is continuous if for each open set U in B,

f^-1(U)

is open in A. A continuous map might “telescope” certain open sets into one another, so that there are fewer open sets in B than there are in A, but as long as each one was mapped to by the elements of an open set in A, we’re OK.

A homeomorphism is simply a continuous map that has a continuous inverse, which means the “undoing map” that puts everything back the way it was is continuous as well.

Example

Let’s use the set S={a, b, c} with the Alexandrov topology {{a, b, c}, {b, c}, {c}, {}}, and the set X={x, y, z}, again with the Alexandrov topology {{x, y, z}, {y, z}, {z}, {}}. The following diagram represents a map f from S to X; each arrow starts at an element of S and points at the element f maps it to:

Is this mapping continuous?

To find out, let’s look at the open sets of X. We have

f^-1({x, y, z}) = {a, b, c}

which is good, because {a, b, c} is open in S. Next,

f^-1({y, z}) = {a, b, c}

which is also good, for the same reason. But here’s a problem:

f^-1({z}) = {a, c}

Now, {a, c} is not an open set in S. That means f fails to be continuous, because to be continuous f^-1(U) must be open in S for every open set U of X.

If we change f so that b maps to z and a maps to y, it becomes continuous, as you can check for yourself:

Inverses of Continuous maps

The map just given doesn’t, however, have an inverse, because there’s no map that “undoes” f. What do we mean by this “undoing”? We mean that if g is the inverse of f then g maps X to S and g(f(q)) is always q itself for every element q in X. Now, we have

f(a)	=y
f(b)	=z
f(c)	=z

Clearly, if g is the inverse of f then g(y) must be a; then g(f(a)) = g(y) = a as required. But for g to be a map from X to S it must assign exactly one element of S to each element of X.

This presents two problems. First, what should g(x) be? There’s no element of S that f maps to x, so it’s not clear. Second, what should g(z) be? both b and c are mapped to z, so which should we choose? Either of these problems would be fatal; they mean we can’t construct an inverse map for f, so f is not invertible.

If you play around with S and X for a while, you’ll find there’s actually only one continuous, invertible map between them, which is this one:

You can check for yourself that this is indeed continuous, it has an inverse and that inverse is continuous as well (this is an exercise well worth doing), so that this map represents a homeomorphism between S and X.

Topological Sameness

When a homeomorphism exists between two spaces we say they’re homeomorphic; and homeomorphic spaces are topologically “the same”. This is what’s meant by the old joke that a topologist can’t tell the difference between a doughnut and a coffee cup — it’s because you can find a homeomorphism between them.

Wikipedia has a nice animation of one such transformation. Remember, the actual map is just from the beginning to the end; that fact that it’s continuous means you can create a “smooth” animation like this out of it, with no jumps or tears.

In the next instalment we’ll look at some examples of properties that can be different between homeomorphic spaces; that is, properties that aren’t topologically relevant.

[UPDATE: There are some more nice animations of homeomorphisms between two-dimensional surfaces here.]

 

This post is a part of a Topology Mini-Series