1. Balls, spheres, fixed points, and retractions
The Dutch mathematician L.E.J. Brouwer (1881–1966) proved some remarkable theorems about ‘continuous’ maps between familiar objects: circle, disk, solid ball, etc. The setting for these was the ‘category of topological spaces and continuous maps.’ For our purposes it is unnecessary to have any precise description of this category; we will instead eventually list certain facts which we will call ‘axioms’ and deduce conclusions from these axioms. Naturally, the axioms will not be selected at random, but will reflect our experience with ‘cohesive sets’ (sets in which it makes sense to speak of closeness of points) and ‘continuous maps.’ (Roughly, a map f is continuous ...