1. Metric spaces. Definition 1.1. Let X be a set. We say is a metric on X if 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 Collections: Mathematics