Schutz: Geometrical Methods of Theoretical Physics 1.1,2

Sections 1.1 and 1.2 are devoted to introducing some ideas of topology needed for the definition of a manifold. R^n is the prototypical manifold, and Schutz sets about defining its open sets with a Euclidean Metric and some variations.

Much or most of modern mathematics can trace its ancestry back to Decartes' marriage of geometry and algebra. The idea of identifying points in space with pairs (or triples, for 3-D) of numbers was the key element, and has a natural generalization to n-tuples of numbers representing points in n dimensional space.

Topology represents space stripped down to an essence - that essence that preserves continuity. What do we mean by continuity anyway, and how do we represent it? One key attribute is that we expect that there should be points arbitrarily close to any other point. Topologists found that they could boil that essence down to the notion of a set and the behavior of its open subsets - but I won't spoil any punchlines here - if you want to know you will have to read the book, or almost any other book on topology or real analysis - or Wikipedia.

Section 1.2 deals with mappings. Legend has it that the Inuit have 50 words for snow. Whether or no, mathematicians have a number of words for function, for reasons similar to the reasons the Inuit are supposed to have all those words for snow - it's that important. Function, map, mapping, transformation, functional, functor are all variations on this theme. A function is in its essence a machine for taking an input from one set and producing a unique corresponding output in another (possibly the same) set.

Geometricians like to use the word map, because the functions of most interest to them take points from one space and associate them to unique corresponding points in another space - just like an ordinary map associates points on a paper (or pixels on a display) with points in the real world.

Schutz takes us through some examples and notations and defines continuous maps and differentiable maps. Continuity makes sense in a topological space, but differentiability will require some more structure, reserved for chapter 2.

The fact that Chapter 1 contains no exercises is a hint that he isn't serious about introducing new ideas yet.

ADDENDUM: The take away from 1.2 is the definition of a continuous map between topological spaces: For topological space M and N, the map f: M->N is continuous at x in M if every open set in N containing
f(x) contains the image of an open set of M containing x.

The pig's mind finds it hard to contain a sentence containing so many instances of the word "contain." I suggest drawing a diagram. It's also useful to convince oneself that this definition is equivalent to the elementary calculus epsilon-delta definition - but doesn't depend on any notion of distance.