BSc Sem V 2015 2016 2016 Math 3 Question Paper - Mumbai University | munotes

Questions asked in this paper

• (2) Figures to the right indicate marks. 6 L

1. Q1 (a) Attempt any one question:
   • i. Let (X,d) be a metric space. Define limit point of F X. Also show that F is closed if and only if F contains all it’s limit points ii, In a metric space (X,d) , prove that arbitrary union of open sets is open in X Give an example to show that arbitrary intersection of open sets is not open
   • (b) Attempt any Two questions: 12
   • i. Define a normed linear space (X, || ||). Show that if (X, ||) is a Space then d: X x X — R defined by y) = — is motri
   • ii. Prove that in any metric space C X, A is closed if and GA where 8A denotes the boundary of A
   • iii. Prove that (N,d) and (N, where d is the usual from R) and is the discrete metric in N, are equivalent metric :
   • iv. Show that A = 5} is both open in the subspace Q of R with usual metric
2. Q2 (a) Attempt any one question: 8 marks
   • i. (X,d) be a metric space, A C x. at pe A if and only if 3 a Sequence A such that conver pin X Define complete metric Space. Prove Space (X, d) is complete if and only closed
   • (b) Attempt any Two questions: Ne)
   • i. Prove that in a discrete metri li. Prove or disprove: Let d : ace is complete ) is bounded in (X then is in Cantors is applicable in the followin a in each ere is a Sequence of find nen x Subsets of R and the distance 12
   • iv. Let be a metric Show t is a convergent Sequences in x for each open G of Y is continuous on i ) 18 an Open subset of x i tinuous at p X if be a function. Show that f is con (tq) in X converging to p, the sequence
   • (b) Attempt any Two questions:
   • i. Let (X,d) be a metric space and f : X R, ( R with usua metric) is fey continuous on X. If > 0 for some X then show that 3 4 > 0 such ii, Let (X,d) and (Y,d’) be metric spaces and f,g: X —> Y be continuo
3. Q10 Show that f(z) = 9(z)} is a closed subset of X iii, Let (X,d) and be metric spaces. When is f : uniformly continuous? Show that f(r) = is uniformly sontinuous on R (under usual metric)
   • iv. Let f be a continuous map. Show that R? defined by = is continuous on R?(Under metric of R and
4. Q4 Attempt any Three questions: 15 marks
   • (b) Let (X,d) be a metric space and A C X. Rw that A is open if and only if
   • A). Hence decide set of rationals is an open subset of R under metric of R
   • (c) Let X = 1] and be the by ll, on X. = i Show that the following functions {f,} is bounded in 64 Let (X,d) be space and AC X Prove that = 0 if and only if
   • (e) —+ (R,d) defined by where metric on R? and + y is continuous on R?,
   • (f) ( and be metric spaces. Show Tic onR on X and if in X is Cauchy then sh X is uniformly chy in Y. ow that the sequence

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