Elements of Metric Spaces — DU B.Sc. (Hons.) Mathematics Previous Year Question Papers

(c) Let the real-valued function f be continuous on the closed and bounded interval [a, b]. Prove that f is bounded and attains its maximum value and minimum value at points of [a, b]. (7.5) : Common Program Group (NEP-UGCF 2022) - GE 4. In the question paper, given notations have their usual meaning unless and until stated otherwise. 1. (a) Define metric on a set. Show that the function d, defined on  by d(x, y) = x y   1 x y for each x, y  , is a metric on . (2+5.5) (b) Prove that in the discrete metric space (X, d) where d denotes the discrete metric, a sequence is convergent if, and only if, it is an eventually constant sequence. Is (X, d) a complete metric space? Justify your answer. (4+1+2.5) (c) Prove that every convergent sequence is a Cauchy sequence but the converse is not true. State one condition under which a Cauchy sequence is convergent. (4+2+1.5) (c) (i) Let (X, d) be a disconnected metric space. Prove that there exist two non-empty disjoint subsets A and B, both open in X with X = (ii) Show that in R, the continuous image of an interval is an interval. (4,3.5) 6. (a) (i) Show that the interval I = (0,1) in the metric space (, d) where d denote the usual metric, is not compact. (ii) Give an example of a non-empty set which is always compact irrespective of the metric imposed on it. (4,3.5) (b) Let Y be closed subset of the compact metric space (X, d). Prove that (Y, dY) is compact. Further show that condition (X, d) is a compact metric space is not redundant. (4+3.5) (c) Let (X, d1) and (Y, d2) be two metric spaces and let f: X  Y. Then show that f is continuous on X if and only if     f A f A  for all subsets A of 5. (a) (i) Given an example of a metric space whose eve ry non-empty non-singlet on set is disconnected. (ii) Does there exist an onto continuous map f: R  R \ {0}? (4,3.5) (b) Let (X, d) be a metric space. Prove that (X, d) is connected if, and only if, the only continuous mappings of (X, d) into (X0, d0) are the constant mappings, where X0 = {0,1} and d0 denotes the discrete metric on X0, (7.5) 2. (a) (i) Let X = , d be the usual metric on . Prove that an open ball in (, d) is a bounded open interval and conversely that any bounded open interval on the real line is an open ball. (ii) Is it possible for a given ball not to have a unique centre or radius? Justify your answer. (4,3.5) (b) Let X = , d be the usual metric and d1 be the discrete metric on R. Let A = {0}        . Determine whether the set A is open or closed or none in the metric spaces (X, d) (c) Let Y be a subspace of a metric space (X, d). Prove that (i) Every subset of Y that is open in Y is also open in X if, and only if, Y is open in X. (ii) Every subset of Y that is closed in Y is also closed in X if, and only if, Y is closed in X. 3. (a) Prove the following : (i) A function is continuous at every isolated point in its domain. (ii) A function with discrete metric space as its domain is continuous.