Let and be topological spaces, and let be a function. We say that is a homeomorphism if
is a bijection, so there is an inverse map , satisfying for all and for all .
Both and are continuous.
We say that and are homeomorphic if there exists a homeomorphism from to . We will write to indicate that and are homeomorphic.
Later we will introduce a different notion called homotopy equivalence, and write (with an extra bar) to indicate that and are homotopy equivalent. It is important to distinguish between these two ideas. However, you should be aware that there is no consistency in the literature about the notation used. It is safest to say “homeomorphic” or “homotopy equivalent” in words, to avoid confusion.
There are very few indirect techniques for proving that two spaces are homeomorphic. Instead, we just have to find a specific homeomorphism. Many examples will be given below.
Suppose that with and . Then and and and .
We can define maps by
We find that and , so is inverse to . It is clear that and are continuous, so is a homeomorphism. The same formulae also give homeomorphisms and so on. ∎
The formulae
give continuous maps which are inverse to each other. Thus, they are both homeomorphisms, and . Moreover, the same maps restrict to give homeomorphisms between and .
First, note that when we have so , so dividing by does not cause any problems. Thus, is a continuous function from to . Similarly, for all we have so , so dividing by does not cause a problem, and is a continuous map from to . Note also that
so , so can in fact be regarded as a continuous map .
Note that if then
so , so . This shows that .
Conversely, suppose we start with and put . We then have
so , so . This shows that , so and are inverses of each other, as required. ∎
Suppose we are trying to define a map , and we give a formula for . For this to be valid, we must check two things.
The formula must be meaningful for all elements . It must never involve division by zero, or square roots of negative numbers in a context where a real result is required, for example.
The resulting value must lie in . For example, might be a subspace of defined by various equations or inequalities. Typically it will only be obvious from the formula that lies in , so we need to check that the equations or inequalities are satisfied as an extra step.
You should observe how these checks were done in the proof of Proposition 4.5. It is sadly common for them to be omitted in homework or exam answers submitted by students. Do not let that be you.
There is some interesting geometry behind the maps and in Proposition 4.5: we have and if and only if and are related as in the diagram below.
In more detail, let be the circle of radius one centred at . Given a point , let be the point on the lower half of lying directly above . Then draw a line from through , and let be the point where it meets the axis. I claim that . Indeed, as the point lies on , the distance from to must be one, so , so . On the other hand, the diagram contains a triangle of base and height , and a nested triangle of base and height . These triangles have the same angles, so we must have , so as claimed.
The previous example can be generalised as follows. Put (so in particular ). We can define and by
Using essentially the same argument as given above, we see that and are mutually inverse homeomorphisms, so is homeomorphic to .
Note that Definition 4.1 specifies that both and must be continuous. It is necessary to make the definition this way, because it can easily happen that we have a bijection where is continuous but is not, and we do not want to count maps like that as homeomorphisms. We give such an example here.
Put and . Define maps by
It is easy to see that and are inverse to each other. We claim that is continuous but that is not. To see this, we introduce the subsets
The sets and are open in , so and are open in the subspace topology on . The map is given by on and by on , so the restrictions to and are both continuous, so is continuous on by open patching (Proposition 3.34). On the other hand, we have , which is not an open subset of the space ; this proves that is not continuous. We therefore have a continuous bijection whose inverse is not continuous, so it does not count as a homeomorphism.
Define by
so
and
It is easy to see that this is a homeomorphism, with inverse
We can also define by
This can also be shown to be a homeomorphism. The inverse is
where
We could try to define a map by the rule
There could be two different things wrong with this definition, depending on how we interpret it.
Any point in can be expressed as for infinitely many different values of , differing by multiples of . For example, the point can be represented as or as . The first representation gives , and the second representation gives . Thus, equation (A) does not give us a well-defined function .
We could instead note that any point in can be expressed as for a unique choice of satisfying the auxiliary condition , and we could define by saying that formula (A) holds for this choice of . This gives a well-defined function, which satisfies . However, for points lying just below , we find that and so is close to . This shows that is discontinuous, despite the apparent continuity of all ingredients in equation (A).
For this kind of reason, it is generally a bad idea to define functions in terms of . This is why our initial definition of in Example 4.11 was given directly in terms of the coordinates , and .
Let be the “north pole” of the sphere , in other words the point . There are mutually inverse continuous maps given by
so is homeomorphic to . This is called stereographic projection. Geometrically, is the unique point where the line joining to meets the plane , and is the unique point where the line joining to meets :
We will leave it to the reader to check directly from the formulae that and . Essentially the same formulae can be used to prove that for any and any .