BSc Mathematics SEM VI 2016 17 2016-17 Metric Topology Question Paper - Mumbai University | munotes

Questions asked in this paper

1. Q3 (a) Attempt any One from the following: 8 marks
   • (i) (X,d) is @ metric space and A, B are subsets of X such that A is connected and BCA. Prove that B is connected. Give an examiple.to. show that if and A,C are connected then B need not be
   • (ii) Prove that a path connected subset, of R” (distance being Euclidean) is connected
   • (b) Attempt any Two from the following: 12
   • (i) If (X,d) be a connected metric space and. (distance in Z being usual distance) is a continuous function then prove that f is a constant function
   • (ii) Show that the set S = 5} is a convex set in (R?, d) where d is the Euclidean distance (X,d) isa connected metric space which is not bounded: Prove that for each 2p X and for each r > 0 the set {z X nonempty
   • (iv) Let A be the union of the following subsets 3 and L-of R2. Show that A is connected (distance in R? being
2. Q4 Attempt any Three from the following: 15 marks
   • (a) f: a continuous such that: f takes only rational values then show that f is
   • (b) Prove that a subspace of (d being usual distance) is not complete but is complete as a subspace of (R, where is discrete metric
   • (c) Let f : > b] is continuous on and differentiable on (a,b). If Je R with that Vx (a,b) then prove that f is a contraction of
   • (d) Let and be metric spaces. that if is uniformly continuous on X and if (an) in X is Cauchy then show that the sequence is Cauchy in Y
   • (e) Show that = > 0, — = 1} is path connected
   • (f) Prove or disprove : If A, B are connected subsets of R with respect to usual distance and ANB then AN B is also connected
   • (a) Attempt any One each open sul 4 (ii) Let then pro\

Read from the scan above, so a character or two may differ. The scan is the original.