BSc Mathematics SEM V ATKT May 2018-19 ATKT MATHEMATICS 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 R —> R is not a metric on R?
   • (c) = (d) d(x, y) = max{2, —
   • ii. Which of the following is a description of the open ball B((0,0),1) for the metric d :
   • (a) It consists of the points within the circle of radius 1 centred at (0,0)
   • (b) It consists of the points inside the square bounded by the lines x+y = 1, = 1,
   • (c) It consists of the points inside the square bounded by the lines y=1,
   • (d) None of the above
   • iii. Let A = {(z,y) x,y Q} and B = [0,1] x [0,1] be subsets of R? with Euclidean
   • (c) x (0,1) (d) None of the above
   • iv. Which of the following sets is not closed in the subspace Q of R (distance being usual)?
   • v. If [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. Which of the following subspaces are dense in R
   • (a) (Z,d), where d is the usual distance. (b) where d is the usual distance
   • (c) where d is the discrete (d) None of these
   • vii. Let (X,d) be a complete metric space , A and B are complete subspaces of (X,d) and B is nonempty then
   • (a) AUB and AN B are complete. (b) AUB is complete and AN B is not
   • (c) AN Bis complete and not. (d) none of the above Which of the following statements is TRUE?
   • (a) is compact in (R,d) where d is usual metric
   • (b) [1,2] is compact in where d; is the discrete metric
   • (c) {1,2,3,4} is a compact set in (N,d), where d is usual metric from R
   • (d) none of these Paper Subject Code: 24246 Mathematics: Topology of Metric Spaces
   • ix. Let A be a compact subset of R. Then
   • (a) A may not be compact. (b) may not be compact
   • (c) 0A may not be compact. (d) None of the above
   • 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) may have subsequences converging to different. limits
   • (d) (a,) is Cauchy
2. Q2 (a) Attempt any One from the following: 8 marks
   • (i) Let (X,d) be a metric space and S X. Show D(S) is a closed subset. of X where D(S) denotes the set of all of S
   • (ii) Let (X,d) be a metric space. Prove the following:
   • (1) Arbitrary union of open sets is open (II) A subset G of X is open if and only if it is an union of open balls
   • (b) Attempt any Two from the following: 12
   • (i) Show that in a discrete metric space (X,d), every subset is both open and closed
   • (ii) Define a metric space (X,d) and give an example of a metric space. Let (X,d) bea metric space, prove that. |d(x,y) — d(x,z)| < d(y, z)
   • (iii) Consider the norms {| and || on R? defined as, for any x = R’, = = and = Show that for
   • (iv) Let (X,d) be a metric space. d, : X x X — R is a metric defined as = Va,y X. Show that are equivalent metrics on X
3. Q3 (a) Attempt any One from the following: 8 marks
   • (i) Let (X,d) space and Y be a non-empty subset of X. Prove that a subset G of Y is open in the subspace (Y,d) if and only if G where V is an open set
   • (ii) Show that [0,1] is uncountable
   • (b) Attempt any Two from the following: 12
   • (i) Check whether Cantor’s Theorem is applicable in each of the following examples and find in each ase, where (F;,) is a sequence of subsets of R and the distance d is usual distance from IR, in each examples:
   • (ii) (0, 1), Fy = [0,
   • (ii) Show that-in a metric sapce (X,d) every Convergent sequence is Cauchy and the converse not. true Paper Subject Code: 24246 Mathematics: Topology of Metric Spaces
   • (iii) Let (X,d) be a metric space and A C X. Show that p A if and only if there is a sequence of points in A converging to p
   • (iv) Show that S = {2 Q: 3 < 2? < 5} is both open and closed in the subspace Q of R with usual metric
4. Q4 (a) Attempt any One from the following: 8 marks
   • (i) Consider the metric space (R,d) where d is usual metric, # R. Prove that if A is closed and bounded then A is sequentially compact
   • (ii) Suppose (X,d) is a metric space and C is a non-empty collection of compact subsets
   • (I) & is acompact subset of (II) If C is finite then K is a compact subset of X
   • (b) Attempt any Two from the following: 12
   • (i) Show that a compact subset of a metric space is closed
   • (ii) Prove or disprove:
   • (I) A compact set in a metric space is not open (II) Interior of a compact set are compact
   • (iii) Consider the metric space = sup {|f(t)| : t Show that the open cover {B(0,n)}nen of Cla, has no finite subcover. (0 being the
   • (iv) Prove that a subset AK in a discrete metric space (X,d) is compact if and only if K is
5. Q5 Attempt any Four from the following: 20 marks
   • (a) Show that every open ball is an open set
   • (b) Let be metrics. on X. Define d: X x X —> Ras d(x,y) = max y), do(z, y)} Show that metric on X and dg be metrics on.a non-empty set such that there exist > 0 such that Let dj and X such that th ist such th < < X. Show that a sequence is Cauchy in if and only if sequence (x,,) is Cauchy in (X,
   • (d) Prove that — +10 = 0 has 4 distinct roots in R
   • (e) Determine which of the following subsets of (R?,d), where d is Euclidean distance is com pact. Justify your answer
   • (ii) D = <1}
   • (f) If A,B are compact subsets of R with respect to usual distance, show that A x B is a compact subset. of R? with Euclidean metric

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