Given two topological spaces, and we say that a map is a homeomorphism if it satisfies the three properties below:
- It is a bijection (that is, both injective and surjective),
- it is continuous, and
- it has a continuous inverse
Two spaces for which there is such a homeomorphism are called homeomorphic. As it turns out, homeomorphic spaces have the same topological properties.
In what follows, we will construct several interesting homeomorphisms to train our skills:
Disks and squares
Consider the unit disk and the unit square defined by
We construct a homeomorphism as follows:
This function maps for each , the border of each square into the circle of radius centered at the origin; the manner in which these sets map into each other indicates why the function is a bijection. It is very easy to check its continuity as well, and I leave that task to the reader.
An inverse to this function is constructed in a similar way, so that each circle is mapped to the border of a square:
An open interval and the real line
Let us find an homeomorphism from the unit interval into the real line based in an interesting construction called stereographic projection. We start by mapping each into an angle by simple multiplication: . This angle gives a single element in the unit circle , except the point The stereographic projection turns each point of the circle different than into a unique point in the plane with coordinates where
This is the function we are looking for. Its continuity is easy to prove, and so are its one-to-one and onto properties. In order to construct the inverse map, trace back from the real line to the vertical line from there to the unit disk by the inverse of stereographic projection through the point , and find the angle of the corresponding image. Division by offers you a value in the unit interval This value is the image of the inverse I leave the construction of the analytical expression of this function to the reader as a nice exercise.