 
Summary: 1. Metric spaces.
Definition 1.1. Let X be a set. We say is a metric on X if
: X × X {r R : r 0}
and
(i) (x, y) = (y, x) whenever x, y X;
(ii) (x, z) (x, y) + (y, z) whenever x, y, z X.
(iii) If x, y X then (x, y) = 0 if an only if x = y.
The inequality in (ii) is called the triangle inequality. A metric space is an
ordered pair (X, ) such that X is a set and is a metric on X.
We now fix a set X and a metric on X.
For each a X and each positive real number r we let
U(a, r) = {x X : (x, a) < r} and we let B(a, r) = {x X : (x, a) r}.
We say a subset U of X is open if for each a U there is a positive real number
such that U(a, ) U. We leave as an exercise to the reader the proof of the
fact that the family of open sets is a topology on X. This topology is called the
topology induced by the metric ; one proves this in exactly the same way we
proved the corresponding fact for Rn
.
Suppose x is a sequence in X and b X. Note that
lim
