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.
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