6 6. Let (X, d) be a metric space and T the topology induced on X by d. Let Y be a subset of X and di the metric on Y obtained by restricting d; that is, di(a, b) d(a, b) for all a and b in Y. If...
Let be a metric space and let be the topology on induced by , and let be a compact space. Prove that is compact. (x, d) We were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this imageAj,i=1,2,... na1 An
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.
(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...
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...
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...
(a) On R2, prove that di ((zı, y), (z2W2)) := Izı-zal + ly,-Val is a metric. (b) Assume that doc ( (zi, yī), (z2,p)) := maxlz-zal, lyi-yl} is a metric on R2 for each p 21. Prove that di and d induce the same topology on R2. You may use the following lemma (but do not need to prove it): Lemma: Let d and d' be two metrics on aset X; let T and T' be the topologies the induce...
Let (X, d) be an infinite discrete metric space. Prove that any infinite subset of X is closed and bounded but NOT compact
Example: Let x, y ∈ Rn, where n ∈ N. The line segment joining x to y is the subset {(1 − t)x + ty : 0 ≤ t ≤ 1 } of R n . A subset A of Rn, where n ∈ N, is called convex if it contains the line segment joining any two of its points. It is easy to check that any convex set is path-connected. (a) Let f : X → Y be an...
9. Let X and Y be metric spaces, and let D be a dense subset of X. (For the definition of "dense, see Problem 4 at the end of Section 3.5.) (a) Let f : X → Y and g : X → Y be continuous functions. Suppose that f(d)gld) for all d E D. Prove that f and g are the same function.
A subset D of a metric space (X, d) is dense if every member of X is a limit of a sequence of elements from D. Suppose (X,d) and (Y,ρ) are metric spaces and D is a dense subset of X. 1. Prove that if f : D -» Y is uniformly continuous then there exists an extension15 of f to a if dn (E D) e X define 7(x) lim f(d,) uniformly continuous function f:X * Y. Hint: 2....