In the following, (X,d) is an arbitrary metric space and (X,d,μ) is an arbitrary metric measure space.
In the following, (X,d) is an arbitrary metric space and (X,d,μ) is an arbitrary metric measure...
In this problem we show that any metric space (X, d) is homeomorphic to a bounded metric space. (a) Define ρ : X X R by Show that ρ defines a metric on X. Conclude that (X,p) is a bounded metric space. (b) Show that f : (X, d) → (X, p) given by f(x) = x is a homeomorphism ism. (c) Is it true that if (X, d) is complete then (X, ρ) is complete? In this problem we...
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....
Problem 1. Let (X, d) be a metric space and t the metric topology on X. (a) Fix a E X. Prove that the map f :(X, T) + R defined by f(x) = d(a, x) is continuous. (b) If {x'n} and {yn} are Cauchy sequences, prove that {d(In, Yn)} is a Cauchy sequence in R.
Using only the definition of compact sets in a metric space, give examples of: (a) A nonempty bounded set in (R", dp), for n > 2 and 1 < pく00, which is not compact. (b) A bounded subset Y of R such that (Y, dy) contains nonempty closed and bounded subsets which are not compact (here dy is the metric inherited from the usual metric in R) Using only the definition of compact sets in a metric space, give examples...
1. (a) Let d be a metric on a non-empty set X. Prove that each of the following are metrics on X: a a + i. d(1)(, y) = kd(x, y), where k >0; [3] ii. dr,y) d(2) (1, y) = [10] 1+ d(,y) The proof of the triangle inequality for d(2) boils down to showing b + > 1fc 1+a 1+b 1+c for all a, b, c > 0 with a +b > c. Proceed as follows to prove...
7. Recall the space m of bounded sequences of real numbers together with the metric d(х, у) — suр |2; — Ук). k 1,2. (a) Give a simple proof to show that m is complete by showing that m = suitable space X. (Recall that C(X) denotes the space of continuous bounded real- valued functions on X together with the supremum norm.) C(X) for some (b) Let A denote the unit ball in m given by А 3 (x€ т:...
6.6.3 Referring to Definition 6.3, prove that (a) A totally bounded metric space is bounded (b) Show by example that there exist bounded metric spaces that are not totally bounded. (c) Consider R" with the Euclidean metric da. Show that a sust ACRis bounded if and only if it is totally bounded.
Let (X,A,μ) be a metric space. 4. Let A and B be two collections of subsets of X. Assume that any set in A belongs to o(B) and that any set in B belongs to O(A). Show that o(A) = 0(B).
(TOPOLOGY) Prove the following using the defintion: Exercise 56. Let (M, d) be a metric space and let k be a positive real number. We have shown that the function dk defined by dx(x, y) = kd(x,y) is a metric on M. Let Me denote M with metric d and let M denote M with metric dk. 1. Let f: Md+Mk be defined by f(x) = r. Show that f is continuous. 2. Let g: Mx + Md be defined...
Let (Q, d) be the metric space consisting of the set Q of rational numbers with the standard metric d(x, y) = |x-yl. Show that the Heine Borel theorem fails for (Q,d). In other words, show that (Q, d) has a subset SCQ that is closed and bounded, but not compact