Let (X,d) be a metric space, SCX, pEX. 1. p is said to be a limit...
Al. Let E be a non-empty set and let d:ExE0, oo). (a) Give the three conditions that d must satisfy to be a metric on E. (b) Ifa E E, r > 0 and 8 0, give the definition of the open ball BE(a) and the closed ball B (a) n-p) closure point of A. Hence, say what it means for A to be a closed subset of E 2 c) Say what it means for a sequence () in...
A point xo in a metric space X is said to be isolated if xo is not in the closure of X x0}. Using the Baire Category theorem, prove that the complement of any countable set is dense in X. Use this to show that an infinite complete metric space with no isolated points is uncountable.
3. (a) Prove the following: Cantor's Intersection Theorem: Let (X, d) be a complete metric space and {Anymore a nested sequence of non-empty closed sets whose diameters D(An) have limit 0. Then An has exactly one member. csc'anno proach onsdelered. c) Show that, in part (a), n A, may be empty if the requirement that the diameters
A. Let (X, d) be a metric space so that for every E X and every r>0 the closed ball N,(z) = {ye X : d(y, z) < r} is com pact. Let be a homeomorphism. (1) Prove that f"-+m-fn。fm for all n, m E Z. (2) Let z E X and suppose that F, {fn (z) : n E 2) is a closed subset of X Prove that F is a discrete subset of X (A subset Y C...
08. (3+2+1+1=7 marks) Let (E, d) be a metric space and let A be a non-empty subset of E. Recall the distance from a point x e E to A is defined by dx, A) = inf da, a).. a. Show that da, A) - dy, A) <d(x,y)Vxy e E. Let U and V be two disjoint and closed subsets of E, and define f: E- dz,U) R by f(x) = 0(2,U) + d(«,V) b. Show that f is continuous...
Let (X, d) be a metric space, and let ACX be a subset (a) (3 pts) Let x E X. Write the definition of d(x, A) (b) (7 pts) Assume A is closed. Prove that d(x,A-0 if and only if x E A. Let (X, d) be a metric space, and let ACX be a subset (a) (3 pts) Let x E X. Write the definition of d(x, A) (b) (7 pts) Assume A is closed. Prove that d(x,A-0 if...
(1) Let (X,d) be a metric space and A, B CX be closed. Prove that A\B and B\A are separated (1) Let (X,d) be a metric space and A, B CX be closed. Prove that A\B and B\A are separated
(5) Here is a fascinating equivalence for being a complete metric space that we will use later. Let (X,d) be a metric space. (b) ** (10 points) Show that the following are equivalent: • (X, d) is complete; • for every family of non-empty closed subsets Fo, F1, F2, ... of X such that F, 2 F12 F22... and limn700 diam( Fn) = 0, it holds that Nnen Fn = {a} for some a € X. (Hint: for the reverse...
Exercise 5 (based on Tao). Let (X,d) be an arbitrary metric space. Prove the following statements (1) If a sequence is convergent in X, all its subsequences are converging to the same limit as the original sequence. (2) If a subsequence of a Cauchy sequence is convergent, then the whole sequence is convergent to the same limit as the subsequence. (3) Suppose that (X,d) is complete and Y S X is closed in (X,d). Then the space (Y,dlyxy) is complete....
(e) Let (X, d) be a metric space and A CX. If x € A is not a cluster point of A, then is a cluster point of Aº. (2 points]