The aim of this note is to give an axiomatic definition of the Brouwer’s topological degree constructed in our previous post.

**Definition:** Let denote the set of tuples where is a bounded open subspace, be a continuous map and . A topological degree is a map satisfying:

*(normality)*if ,*(additivity)*Let be two disjoint open subspaces such that . Then

.

*(homotopy invariance)*Let be a homotopy and be a continuous map satisfying for all . Then

.

First, let us consider some consequences of the definition:

**Property 1:** Let be a topological degree. Then

- If then there exists such that
- For all , ,
- For all , ,
- is constant on any connected component of ,
- Let be a continuous map and satisfying

.

Then .

**Proof.** *First point:* Suppose by contradiction that . Then hence

by additivity. But hence

by additivity, ie. . We deduce that .

*Second point:* Let and . Notice that for some implies . Therefore, by homotopy invariance:

,

that is

*Fourth point:* The fifth point shows that is locally constant. So it is sufficient to conclude by connectedness.

*Fifth point:* Let and . Suppose by contradiction that there exist and such that , that is

.

Then

,

a contradiction. Therefore, by homotopy invariance

.

*Third point: *Let ; the map is well-defined since .

We shall prove that is locally constant at . Then, we will be able to deduce that is locally constant at any point from the equality

and so constant by connectedness of , hence

.

First, let and suppose by contradiction that for every .

By compactness of , there exists a decreasing sequence converging to so that converge to some . So

,

that is . Thus , a contradiction. So let so that .

In particular, hence by additivity

.

Afterwards, if then , hence

.

Therefore, because , hence by additivity

For fixed, let be a homotopy between and . Notice that for all and ,

,

so ; consequently, since , and by homotopy invariance

We deduce from , and that for all

Now we are able to prove our main theorem:

**Theorem 2:** There exists only one topological degree .

**Proof.** The existence was proved in our previous post, so it is sufficient to show the uniqueness. In order to make the proof clear, let us proceed step by step.

**Step 1:**Let .

According to the last point of our property and to Stone-Weierstrass approximation theorem, we may suppose that is in fact on . Then, according to the last point of our property and to Sard’s lemma, we may suppose that is a regular value of .

**Step 2:**Let where is on and is a regular value of .

According to the inverse function theorem, is a local diffeomorphism at any point of . Therefore, is a closed discrete subset of the compact , so is finite.

Let and be such that and . By additivity and our previous property:

Notice that the previous equality still holds if we replace with any element of .

Let be fixed. Suppose by contradiction that for any , the homotopy between and has a zero . Because

we deduce that . On the other hand, is invertible so there exists such that for all satisfying . Therefore, , hence a contradiction since does not depend on .

By homotopy invariance, we get

,

for some . Therefore we may suppose that is a linear isomorphism, has the form and .

**Step 3:**Let where is a linear isomorphism.

By polar decomposition, there exist and such that .

In particular, there exist and such that

and there exist and such that

For all , let

and

Using , we can define a homotopy of isomorphisms between and

hence

If then and . From now on, suppose .

Without loss of generality, by additivity we may replace with in the coordinates associated to . Let

and

Let be a homotopy between and . Notice that:

- If then the first coordinate of is ,
- If then the first coordinate of is ,
- If with then the th coordinate of is .

Therefore, for all hence by homotopy invariance. Moreover, for all so .

Then by additivity

,

since and coincide on , and

Therefore, we get

Finally, we deduce that .

**Corollary:** A topological degree has integer values.