BSc Mathematics SEM V 2018 19 2018-19 Maths Topology Of Metric Spaces Question Paper - Mumbai University | munotes

Questions asked in this paper

• (2) Figures to the right indicate marks

1. Q1 Choose correct alternative in each of the following: 20 marks
   • i. Which of the following maps d: R? x R? —> R is a metric on R?, where d is defined as,
   • ii. Consider the discrete metric d defined on a non-empty subset X containing at least two elements. Then, for p X, which of the following is false?
   • iii. Which of the following sets is bounded in R?
   • (a) A =R with respect to the usual metric
   • (b) with respect to the discrete metric
   • (c) A= |a,co) with respect to the usual metric
   • (d) None of the above
   • iv. Let A=Q, B= +) be subsets of R(distance being usual),
   • v. f : [0,1] > [0,1] is defined by f(x) = , then
   • (a) f is continuous on and does not. satisfy intermediate value property
   • (b) f satisfies intermediate value property but. f is not continuous
   • (c) f is continuous only at and = 1]
   • (d) none of the above
   • vi. Every Cauchy Sequence is eventually a constant in
   • (a) (N,d) , where d is the usual distance. (b) (Q,d) , where d is the usual distance
   • (c) (R\ , where dis the usual distance. (d) None of the above
   • vii. Let (X,d) be a complete metric space , A and B are complete subspaces of (X,d) and AN B is nonempty then
   • (a) AUB is complete and AN B is not. (b) B is complete and AU B is not
   • (c) AUB and AN B are complete. (d) none of the above
   • viii. In (R,d), where d is usual metric,
   • (a) is compact. (b) {1} is not compact
   • (c) is compact. (d) None of these
   • ix. In the metric space (Z,d), (Z is the set of integers, d is usual distance), K C Z is compact
   • (a) if and only if closed. (b) if and only if kK is bounded
   • (c) if and only if A has a limit point. (d) if and only K Paper Subject Code: 24246 Mathematics: Topology of Metric Spaces
   • x. Let be a sequence in with usual metric from R. Then, which of the following is
   • (a) has a convergent subsequence
   • (b) is bounded but may not be convergent
   • (c) is Cauchy
   • (d) may have subsequences converging to different limits
2. Q2 (a) Attempt any One of the following: 8 marks
   • (i) Define an open ball in a metric space (X,d) and show that every open ball is an open set. Also give an example to show that the converse need not be true
   • (ii) Let (X,d) be a metric space and A,B C X. Show that (II) = (III) C and the inequality may be strict
   • (b) Attempt any Two of the following: 12
   • (i) State and prove Hausdorff property in a metric space (X,d)
   • (ii) Let (X,d) be a metric space and A C X. Show that A is a closed set and it is the
   • (iii) Prove that (Z,d) and where d is the usual distance (induced from R ) and d; is the discrete metric in Z, are equivalent metric spaces
   • (iv) Answer the following:
   • (I) Consider the subspace [0,00) of IR where distance d in R is usual. Find an open ball in the subspace (Y, d) (II) In (R, usual), show that Q is neither a closed set nor an open set
3. Q3 (a) Attempt any One of the following: 8 marks
   • (i) State and prove Nested interval theorem in R
   • (ii) Let (X,d) be a metric space and A be a subset of X. Show that p X is a limit point of A if and only if there is a sequence of distinct points in A converging to p
   • (b) Attempt any Two of the following: 12
   • (i) Let (X,d) be a metric space and be a Cauchy sequence in X. If has a convergent subsequence then prove that sequence itself is convergent
   • (ii) Let (X,d) be a metric space and Y be a non-empty subset of X. Prove that a subset G of Y is open in the subspace if and only if G = VOY where V is an open set
   • (iii) Prove metric space (R?,d,) is complete where the metric is given by
   • (iv) If f : —> Ris a continuous such that f takes only rational values then show that f is a constant function Paper Subject Code: 24246 Mathematics: Topology of Metric Spaces
4. Q4 (a) Attempt any One of the following: 8 marks
   • (i) Show that a compact subset of a metric space is closed and bounded. Give an example to show that a closed and bounded subset need not be compact
   • (ii) Consider the metric space (R,d) where d is usual metric, @ 4 A Prove that if A is closed and bounded then A satisfies Hein-Borel property
   • (b) Attempt any Two of the following: 12
   • (i) Let A,B be compact subsets of a metric space (X,d). Show that AU B and ANB are compact subsets of (X, d)
   • (ii) Prove that a closed subset of a compact metric space is compact
   • (iii) Consider the metric space d), where dis the usual distance. Show that {(4, 1) N} is an open cover of (0,1). Is (0,1) compact? Justify your answer
   • (iv) Prove or disprove:
   • (1) Interior of a compact set is compact (II) A closed ball in a metric space is compact
5. Q5 Attempt any Four of the following: 20 marks
   • (a) || and || are norms on R? where for = R?, = Show that < and < V2 for R?
   • (b) Define distance of a point p from set a metric space (X,d). If A C X then show that
   • (c) Prove or disprove: Let be equivalent metrics on a non-empty set X. If is bounded in then is bounded in (X,
   • (d) Prove that in a discrete metric space every Cauchy sequence is eventually constant. Hence deduce that a discrete metric space is complete (ce) Consider the set K = Q in a metric space (Q,d) where d is a usual metric from R. Is the set AK compact in (Q,d)? Justify your answer
   • (f) Prove that a subset of a discrete metric space is compact if and only if it is finite

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