22 1. From the Real Numbers to the Complex Numbers

3) δ(x,y) = δ(y,x) for all x,y ∈ X (the distance from x to y is the same as the

distance from y to x).

4) δ(x,w) ≤ δ(x,y) + δ(y,w) for all x,y,w ∈ X (the triangle inequality).

Deﬁnition 6.2. Let X be a set, and let δ be a distance function on X. The pair

(X,δ) is called a metric space.

Mathematicians sometimes write that “the distance function δ makes the set

X into a metric space”. That statement is not completely precise. It ignores

a somewhat pedantic point: the set X is not the metric space; a metric space

consists of the set and the distance function. Many diﬀerent distance functions are

possible for most sets X.

Sometimes we use the word metric on its own to mean the distance function

on a metric space. The most intuitive example is the real number system; putting

δ(x,y) = |x - y| makes R into a metric space. There are many other possibilities

for metrics on R; for example, putting δ(x,y) equal to 1 whenever x 6 = y gives

another possible distance function. In Chapter 2 we formally deﬁne the absolute

value function for complex numbers. Then C becomes a metric space when we put

δ(z,w) = |z - w|. Again, many other distance functions exist. We mention these

ideas now for primarily one reason. The basic concepts involving sequences and

limits can be developed in the metric space setting. The concept of completeness

is then based upon Cauchy sequences; a metric space (X,δ) is complete if and only

if every Cauchy sequence in X has a limit in X. Given a metric space that is not

complete in this sense, it is possible to enlarge it by including all limits of Cauchy

sequences. This approach provides one method for deﬁning the real numbers in

terms of the rational numbers. Recall that our development assumes the existence

of the real numbers and deﬁnes the rational numbers as a particular subset of the

real numbers.

In a metric space there are notions of open and closed balls. Given p ∈ X,

we write Br(p) for the set of points whose distance to p is less than r; we call

Br(p) the open ball of radius r about p. The closed ball also includes points whose

distance to p equals r. Depending on the metric δ, these sets might not resemble

our usual geometric picture of balls. A subset S of a metric space is open if, for

each p ∈ S, there is an 0 such that B (p) ⊂ S. In most situations what counts

is the collection of open subsets of a metric space. It is possible and also appealing

to deﬁne all the basic concepts (limit, continuous function, bounded, compact,

connected, etc.) in terms of the collection of open sets. The resulting subject is

called point set topology. In order to keep this book at an elementary level, we will

not do so; instead we rely on the metric in C deﬁned by δ(z,w) = |z - w|. The

distance between points in this metric equals the usual Euclidean distance between

them.

Occasionally we will require that an open set be connected, and hence we give

the deﬁnition here. In Deﬁnition 6.3 and a few other times in this book, the symbol

∅ denotes the empty set. The empty set is open; there are no points in ∅, and

hence the deﬁnition of open is satisﬁed, albeit vacuously.