Some of the concepts with which we have been concerned are not special to the real or complex numbers but are connected to the general idea of distance, in particular the notion of objects being very close to each other. These ideas can be made general and precise. The abstract concept allows us to develop once and for all a vocabulary and basic results that apply in many circumstances beyond the real numbers. In particular, there are various useful senses in which functions might be considered as being very close to each other (or not).

Metrics

The abstract setting for the notion of distance is a set S, whose elements may be referred to as points. A distance function or metric on S is a function that assigns to each pair of points p and q in S a real number d(p, q) and has the properties, for all p, q, and r in S:

1. D1 (Positivity) d(p, q) ≥ 0; d(p, q) = 0 if and only if p = q.

2. D2 (Symmetry) d(p, q) = d(q, p).

3. D3 (Triangle inequality) d(p, r) ≤ d(p, q) + d(q, r).

Definition. A metric space is a pair (S, d), where d is a metric defined on the set S.

Examples

1. If S is an arbitrary set, the discrete metric on S is defined by d(p, p) = 0, while d(p, q) = 1 if q ≠ p.

2. Suppose that S consists of all strings of length n of 0's and 1's; two such strings of length 5 are 00101 and 10110. Let d(p, q) be the number of places in which the strings differ; for our example, d(p, q) = 3. (If q is supposed to be a copy of p, d(p, q) is the number of errors.)