Answer Happy

VEA > Question 1 Let A be a non-empty subset of M. For x € M, set d(x, A) = inf d(x,y)= da(x) and call this the distance from r to A. (a) Let € M. Show that I ead?, A) = 0. (b) Let e > 0 and set B(A, €) = {r € M: d(3, A) < £}. Show that A = n B(A.€). (c) Let A and B be two nonempty subsets of M such that dA= db. Using the fact that ds = dġ for all nonempty subsets S of M, show that Ā= B. (15, 15, 10 Marks) Question 2 Let f: R+ + R denote a map that is increasing and satisfies f(0) = 0. Suppose moreover that is sub-additive, i.e. f satisfies f(u + v) < $(u) + f(v), Wu,ve Rt. Then the distance function d' = f(d) is also a metric on M. Here one has d'(x, y) = f(d(x,y)], for all x, y e M. (a) Show that the map /: R+ +R: f(x) = min {1, x} is sub-additive. (b) Show that D= min{1,d} defines a metric on M. [ 15, 10 Marks) Question 3 Recall that the metric space (M. d) is separated, in the sense that for all my e M there are two open sets U and V such that I EU, YEV, and U NV = . Let : E M and B a nonempty subset of M such that 2 € B. (a) Can one find two open sets U and V such that X EU, B CV and UV = 0? Explain. (b) Suppose now that r¢ B an set : = d(x,B) := infyeBd(x,y). Show that U = B(x, ) and V = Uyes Bly, ) are two open sets separating r and B. That is, I EU, BCV, and UNV = 0. [ 10, 25 Marks]