This section provides a thumbnail sketch of those elements of point-set topology (also called general topology or just plain topology) that are used in this book. The subject grew out of attempts to rid the notion of continuity of its traditional dependence on the notion of distance. It turned out that continuity could be defined without using real numbers at all; the subject could be founded, instead, on the calculus of sets. Unfamiliarity with the latter is perhaps the main source of difficulty for the beginner.

Detailed treatments of the material discussed below may be found in standard textbooks such as (Kelley, 1955, Willard, 1970 and Munkres, 1975). Of these, the one by Munkres will perhaps be the easiest for the physicist.

Topological spaces

Point-set topology (usually called topology for short) may be regarded as the study of the notions of convergence of sequences1 and continuity of maps without using the notion of real numbers. In the theory of functions of a real variable, both of these notions are intimately related to that of neighbourhoods of a point. A neighbourhood of a point x on the real line is any subset that contains an open interval (x − a, x + a) around x, where a > 0; usually a is a small number, but it does not have to be.