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...
(b) Find an example of an open set G in a metric space X and a closed subset F of G such that there is no δ > 0 with {x : dist(x, F) < δ} C G
Q7 Let A, B C M where M is metric space. Suppose there exist open sets , V C M such that A C B C V and V-0. Prove that A and B are separated.
Q7 Let A, B C M where M is metric space. Suppose there exist open sets , V C M such that A C B C V and V-0. Prove that A and B are separated.
New problems for 2020 1. A topological space is called a T3.space if it is a T, space and for every pair («,F), where € X and F(carefull), there is a continuous function 9 :X (0,1 such that f(x) 0 and f =1 on F. Prove that such a space has the Hausdorff Separation Property. (Hint: One point subsets are closed.] 2. Let X be topological space, and assume that both V and W are subbases for the topology. Show...
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).
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 M be a ơ-algebra of subsets of X and P the set of finite measures on M. Prove that (a) d(μ, v) = supAEM |μ( A)-v(A)| defines a metric on P (b) (P, d) is complete.
Let M be a ơ-algebra of subsets of X and P the set of finite measures on M. Prove that (a) d(μ, v) = supAEM |μ( A)-v(A)| defines a metric on P (b) (P, d) is complete.
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...
(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...
Timelimit Total Points Possible: 19 How many subsets does the set D = {c, a, t} have? Questions Q1 Q2 Q3 Q4 Q5 Q6 Q7 Q8 Q9 Q10 Q11 Q12 Q 13 Points possible: 1 This is attempt 1 of 1 Submit Q14 Q15 Q16 Q17 Q 18 Q19 Print Version 44 11 Feo F10 % & 2 3 4 7 9 W R E T Y U P O * CO est Timelimit: 1 hour, 15 minutes. 1: Which...