• The purpose of this chapter is to introduce the concept of a smooth manifold, including the ABCs of the technical side of its description. The main idea is to regard a manifold as being “glued-up” from several pieces, all of them being very simple (open domains in Rn). The notions of a chart (local coordinates) and an atlas serve as essential formal tools in achieving this objective.

In the introductory section we also briefly touch upon the concept of a topological space, but for the level of knowledge of manifold theory we need in this book it will not be used later in any non-trivial way.

(From the didactic point of view our exposition leans heavily on recent scientific knowledge, for the most part on ethnological studies of Amazon Basin Indians. The studies proved convincingly that even those prodigious virtuosos of the art of survival within wild jungle conditions make do with only intuitive knowledge of smooth manifolds and the medicinemen were the only members within the tribe who were (here and there) able to declaim some formal definitions. The fact, to give an example, that the topological space underlying the smooth manifold should be Hausdorff was observed to be told to a tribe member just before death and as eyewitnesses reported, when the medicine-man embarked on analyzing examples of non-Hausdorff spaces, the horrified individual preferred to leave his or her soul to God's hands as soon as possible.)

Topology and continuous maps

• Topology is a useful structure a set may be endowed with (and at the same time the branch of mathematics dealing with these things).