Topology of Plane Sets of Points 91(S, d) consisting of a nonempty set S and a distance d defined on S x S, ·i.e., defined for every pair of elements in S. On the same set Sit is possible
to define different distances d, d', d", ... leading to different metric spaces
(S, d), (S, d'), ... , as we have seen in the case of the set of complex numbers.
Although at the beginning of our treatment of complex analysis our
interest will be restricted to sets of complex numbers endowed with either
of the first two types of distances mentioned above, later developments
require the consideration of other types of spaces. For this reason in this
chapter we consider the general theory of metric spaces, as well as some
notions concerning topological spaces. There is an additional advantage in
discussing the abstract theory because in this general setting the essence
of the principles involved are more clearly understood.
The general concept of metric space was introduced by Frechet in 1906
[7].
Definition 2. 7 A metric space is a pair ( S, d), where Sis a nonempty set
and dis a distance or metric on S, i.e., a real-valued function defined on
S x S satisfying the following conditions:- d(x, y) ~ 0 for every x E S and every y E S
- d(x, y) = 0 iff x = y
- d(x, y) = d(y, x)
- d(x, z) ~ d(x, y) + d(y, z), x, y, z E S
Condition 3 is called the axiom of symmetry, and condition 4 is called
the triangle axiom (also the triangular law of distances or, more simply,
the triangle inequality). When no misunderstanding is likely, the metric
space (S, d) is denoted by S.
Examples 1. Let S be any nonempty set, and let
d(x, y) = { ~if x = y
if x f:. y
All four conditions above are easily verified, so that (S, d) is a metric space.
This is a rather trivial example, but it shows that any nonempty set can
be made into a metric space.
- The set of the real numbers with distance function defined by
d(x,y) =Ix~ YI
is a metric space, denoted IR^1 •
- The set of the complex numbers with distance function defined by
d(z, z') = iz -z'I
is a metric space. We shall denote this space by (C, d), or by C for short.