Metric SpacesRead pages 23-32Def. A metric space(X , d) is a set X together with a distancefunction d: X × X → R satisfying, for all x, y, and z in X :d(x, y ) ≥ 0, d(x, y )=0 if and only if x = yd(x, y )=d(y, x)d(x, z) ≤ d(x, y)+d(y, z)Note: The last inequality is called the Triangle Inequality andcan also be written as d(x, y) ≥ d(x, z) − d(y, z).Mth 430 – Winter 2006Metric Spaces1/6Examples(Rn, d)Subsets of Rnwith the metric from RnAny set, discrete metricOther metrics on RnHilbert Space (Square summable sequences)Read details on pages 24–25.Mth 430 – Winter 2006Metric Spaces2/6Product MetricTheorem: Let (X1, d1) and (X2, d2) be metric spaces. Then thefollowing is a metric on X1× X2called the product metric:d((a1, a2), (b1, b2)) =(d1(a1, b1))2+(d2(a2, b2))2Proof:Examples:Mth 430 – Winter 2006Metric Spaces3/6Open Sets in Metric SpacesDef. If (X , d) is a metric space, the open ball of radius r > 0about a point p∈ X , Br(p), is defined to be{x ∈ X|d(x, p) < r}Def. If (X, d) is a metric space, a subset U of X is said to beopen in X if∀p ∈ U, ∃εp> 0 so that Bεp(p) ⊂ U.Lemma: Open balls in(X , d) are open.Examples:Mth 430 – Winter 2006Metric Spaces4/6Properties of Open Sets in Metric SpacesTheorem: Let (X , d) be a metric space.X and/0are openAny union of open sets is openAny finite intersection of open sets is open.Proof:Mth 430 – Winter 2006Metric Spaces5/6Continuity in Metric SpacesDef. Let (X , d) and (Y , d) be metric spaces. A functionf: X → Y is continuous at p ∈ X if for each ε > 0 there is aδ > 0 so that d(f (p), f (x)) < ε whenever d(p, x) < δ.Def. Let(X , d) and (Y , d) be metric spaces. A functionf: X → Y is continuous if it is continuous at each point of X .Theorem: Let(X , d) and (Y , d) be metric spaces. A functionf: X → Y is continuous if and only if f−1(U) is open in Xwhenever U is open in Y .Corollary: Compositions of continuous functions arecontinuous.Mth 430 – Winter 2006Metric