Topological Degree in Finite Dimensions

Abstract

In this basic chapter we shall study some basic problems concerning equations of the form f (x) = y, where f is a continuous map from a subset Ω ⊂ ℝⁿ into ℝⁿ and y is a given point in ℝⁿ. First of all we want to know whether such an equation has at least one solution x ∈Ω. If this is the case for some equation, we are then interested in the question of whether this solution is unique or not. We then also want to decide how the solutions are distributed in Ω. Once we have some answers for a particular equation, we need also to study whether these answers remain the same or change drastically if we change f and y in some way. It is most probable that you have already been confronted, more or less explicitly, by all these questions at this stage in your mathematical development.

Everything should be made as simple as possible, but not simpler.

Albert Einstein

When a mathematician has no more ideas, he pursues axiomatics.

Felix Klein

I hope, good luck lies in odd numbers...

They say, there is divinity in odd numbers, either in nativity, chance, or death.

William Shakespeare