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Review of Metric Spaces Lecture 2: Review of Metric Spaces Hart Smith Department of Mathematics University of Washington, Seattle Math 524, Autumn 2013 Hart Smith Math 524 Definition of a Metric Space A metric space consists of: a set X , and function (metric) ρ : X × X ! [0; 1) ; such that: ρ(x; y) = ρ(y; x) (Symmetry) ρ(x; y) = 0 iff x = y (Non-degeneracy) ρ(x; y) + ρ(y; z) ≤ ρ(x; z) (Triangle inequality) Examples of metrics on Rn: n 1=2 X 2 Euclidean metric: ρ(x; y) = jxj − yj j j=1 Box metric: ρ(x; y) = max jxj − yj j j Hart Smith Math 524 Open sets in a metric space A subset O ⊂ X is open if: for each x 2 O there exists δ > 0 ( δ can depend on x ) such that y 2 O whenever ρ(x; y) < δ : B(z; r) ≡ x : ρ(x; z) < r g is open, by triangle inequality. The sets X and ; are both open. The union of any collection of open sets is open. The intersection of a finite collection of open sets is open. The collection of open subsets of X is a topology on X. Hart Smith Math 524 Closed sets in a metric space A subset F ⊂ X is closed if: the complement F c ≡ XnF is open. The intersection of any collection of closed sets is closed. The union of a finite collection of closed sets is closed. For any set E ⊂ X, define the interior and the closure of E: Interior = largest open set contained in E: Eo = int(E) = [ O : O ⊂ E is open Closure = smallest closed set containing E: E = \ F : F ⊃ E is closed Hart Smith Math 524 Sequences in a metric space 1 A sequence fxng ⊂ X converges to x if lim ρ(xn; x) = 0 : n=1 n!1 1 A sequence fxng ⊂ X is Cauchy if lim ρ(xm; xn) = 0 : n=1 m;n!1 8 > 0 ; 9 N < 1 ; such that ρ(xm; xn) < if m; n > N : Every convergent sequence is Cauchy 1 The point x is a cluster point of the sequence fxngn=1 if, for every r > 0, B(x; r) contains xn for infinitely many n. 1 If x is a cluster point of fxngn=1 then some sub-sequence 1 of fxngn=1 converges to x. If a Cauchy sequence has a cluster point x, then the sequence converges to x. Hart Smith Math 524 Complete metric spaces A metric space (X; ρ) is complete if: 1 each Cauchy sequence fxngn=1 ⊂ X converges to some x 2 X. A closed subset E ⊂ X of a complete metric space is complete; i.e. every Cauchy sequence contained in E converges to a point in E. R with Euclidean distance is complete: let x = lim inf xn n R with Euclidean distance is complete: let xjj = lim inf xnjj 1 Rnf0g with Euclidean distance is not complete: xn = n A complete metric space has no holes in it. Hart Smith Math 524 Compact sets in a metric space For E ⊂ X, an open cover of E is: a collection Oα α2A of open subsets such that E ⊂ [α2AOα : A subset E ⊂ X is compact if: every open cover Oα α2A of E has some finite sub-collection N Oj j=1 that covers F. 1 1 Open interval (0; 1) ⊂ R is not compact: On = ( n ; 1) n=1 : A compact set is closed: suppose E compact and x 2= E. 1 Let On = y : ρ(y; x) > n . Finite cover n ≤ N means 1 ρ(y; x) ≥ N for all y 2 E, so x 2= E ; ) E = E : Hart Smith Math 524 Sequentially compact sets A subset E ⊂ X is sequentially compact if: 1 every sequence fxngn=1 ⊂ E has a cluster point in E. E sequentially compact ) E complete if a Cauchy sequence has a cluster point then it converges. Theorem If E ⊂ X is compact then E is sequentially compact. Proof. 1 By contradiction: if no point in E is a cluster point of fxngn=1 then each x has some rx > 0 so that B(x; rx ) contains xn for at most finitely many n. Some finite number cover E, )( Hart Smith Math 524 Totally bounded sets A subset E ⊂ X is totally bounded if: for every r > 0, E is covered by a finite collection of r-balls: N N E ⊂ [n=1B(xn; r) for some finite collection fxngn=1 ⊂ E. A compact set E is totally bounded. For subsets E ⊂ Rn, a set is totally bounded if and only if it is contained in B(0; R) for some R < 1. We have shown: A compact set E is sequentially compact. A compact set E is complete, and it is totally bounded. Hart Smith Math 524 Theorem For subsets E of a metric space, the following are equivalent: (i) E is compact. (ii) E is sequentially compact. (iii) E is complete and totally bounded. Scheme of proof: Have shown (i) ) (ii) and (i) ) (iii). Will show (iii) , (ii), and then (iii)& (ii) ) (i). Hart Smith Math 524 Complete & totally bounded ) sequentially compact 1 Given sequence fxngn=1 ⊂ E, need construct cluster point x. −j Idea: if yj ! x, and B(yj ; 2 ) contains infinitely many elements −j of fxng, then x is a cluster point: B(x; r) ⊃ B(yj ; 2 ) if j large. Take finite cover of E by 2−1 balls: −1 9 y1 2 E : B(y1; 2 ) contains infinitely many elements of fxng. −1 −2 Take finite cover of B(y1; 2 ) \ E by 2 balls: −1 −2 9 y2 2 B(y1; 2 ) \ E : B(y2; 2 ) contains infinitely many fxng. −j −j−1 9 yj+1 2 B(yj ; 2 ): B(yj+1; 2 ) ⊃ infinitely many fxng. −j The sequence fyj g is Cauchy since ρ(yj+1; yj ) ≤ 2 . By completeness of E, yj ! x for some x 2 E. Hart Smith Math 524 Sequentially compact ) complete & totally bounded Sequentially compact ) complete is easy: If Cauchy fxng ⊂ E has a cluster point x 2 E, it converges to x. Not totally bounded ) not sequentially compact: n If not totally bounded, 9 r > 0 : E 6⊂ [j=1B(xj ; r) for any fxj g. Choose x1 2 E Choose x2 2 E : x2 2= B(x1; r) ··· n Choose xn 2 E : xn+1 2= [j=1B(xj ; r) 1 Result: a sequence fxngn=1 such that ρ(xm; xn) > r for all m; n; so it cannot have a cluster point . Hart Smith Math 524 Sequentially compact & totally bounded ) compact Claim: if E is sequentially compact, and E ⊂ [αOα then there exists r > 0 : 8x 2 E ; B(x; r) ⊂ Oα for some α. −n Suppose not: take xn 2 E such that B(xn; 2 ) 6⊂ Oα for any α. 1 The sequence fxngn=1 has a cluster point x 2 E. For some α ; x 2 Oα ; so B(x; r) ⊂ Oα some r > 0. r r Take n such that 2−n < ; and ρ(x ; x) < . 2 n 2 −n Then B(xn; 2 ) ⊂ B(x; r) ⊂ Oα ; )( Hart Smith Math 524 Remarks Compactness of E ⊂ X is a topological property: it depends only on the collection of open subsets of X. Let (X1; ρ1) and (X2; ρ2) be metric spaces If F : X1 $ X2 is a 1-1, onto mapping of sets, and both F and F −1 map opens sets to open sets, then F and F −1 map compact sets to compact sets. Total boundedness & completeness depend on the metric: π π 1-1, onto Consider the map: tan x :(− 2 ; 2 ): ! R open intervals $ open intervals ) open sets $ open sets. π π (− 2 ; 2 ) is totally bounded, not complete (Euclidean metric) R is complete, but not totally bounded. Hart Smith Math 524 Equivalence of metrics Definition Two metrics ρ1 and ρ2 on X are equivalent if there is C > 0: ρ1(x; y) ≤ C ρ2(x; y) ; ρ2(x; y) ≤ C ρ1(x; y) : All basic metric space notions are equivalent for ρ1 for ρ2 : Cauchy sequence, completeness, total boundedness, ::: The metrics ρ1(x; y) = j tan x − tan yj ; ρ2(x; y) = jx − yj π π on (− 2 ; 2 ) are not equivalent, but give the same collection of open sets. Hart Smith Math 524.
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