OSU MTH 430 - Metric Spaces (2 pages)

Examples Metric Spaces Rn d Read pages 23 32 Subsets of Rn with the metric from Rn Def A metric space X d is a set X together with a distance function d X X R satisfying for all x y and z in X Any set discrete metric Other metrics on Rn d x y 0 d x y 0 if and only if x y Hilbert Space Square summable sequences d x y d y x d x z d x y d y z Read details on pages 24 25 Note The last inequality is called the Triangle Inequality and can also be written as d x y d x z d y z Mth 430 Winter 2006 Metric Spaces 1 6 Mth 430 Winter 2006 Metric Spaces 2 6 Open Sets in Metric Spaces Product Metric Theorem Let X1 d1 and X2 d2 be metric spaces Then the following is a metric on X1 X2 called the product metric d a1 a2 b1 b2 Def If X d is a metric space the open ball of radius r 0 about a point p X Br p is de ned to be x X d x p r d1 a1 b1 2 d2 a2 b2 2 Proof Def If X d is a metric space a subset U of X is said to be open in X if p U p 0 so that B p p U Examples Lemma Open balls in X d are open Examples Mth 430 Winter 2006 Metric Spaces 3 6 Mth 430 Winter 2006 Metric Spaces 4 6 Properties of Open Sets in Metric Spaces Continuity in Metric Spaces Def Let X d and Y d be metric spaces A function f X Y is continuous at p X if for each 0 there is a 0 so that d f p f x whenever d p x Theorem Let X d be a metric space X and 0 are open Any union of open sets is open Def Let X d and Y d be metric spaces A function f X Y is continuous if it is continuous at each point of X Any nite intersection of open sets is open Proof Theorem Let X d and Y d be metric spaces A function f X Y is continuous if and only if f 1 U is open in X whenever U is open in Y Corollary Compositions of continuous functions are continuous Mth 430 Winter 2006 Metric Spaces 5 6 Mth 430 Winter 2006 Metric Spaces 6 6