BSc Sem V 2023 2024 Nov 2024 MATHEMATICS TOPOLOGY OF METRIC SPACES Question Paper - Mumbai University | munotes

Questions asked in this paper

• (2) Figures to the right indicate marks

1. Q1 (a) Attempt any One from 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) Show that for a subset of a metric space (X,d), the following statements are equiv
   • (1) F is closed (II) F contains all its limit points
   • (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) be a metric space, prove that y) — d(x, z)| < d(y, z)
   • (iii) Let (X,d) be a metric space and A C X. Show that A is open if and only if A = (Interior of A). Hence decide whether the set of rationals is an open subset of R under usual metric of R
2. Q2 (a) Attempt any One from the following: 8 marks
   • (i) State and prove Cantor’s Intersection Theorem for a metric space (X, d)
   • (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 (Y, d) if and only if G = V NY where V is an open set
   • (b) Attempt any Two from the following: 12
   • (i) Let (X,d) be a metric space and (z,,) be a Cauchy sequence in X. If has a convergent subsequence then prove that sequence (z,,) itself is convergent
   • (ii) Show that A = {x Q: —V2 < x < V2} is both open and closed in the subspace Q of R with usual distance
   • (iii) Let R be such that < y. Show that there exists a number r Q such that
3. Q3 (a) Attempt any One from 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 of a metric space need not be compact
   • (ii) Consider a metric space where d is usual metric, # A C R. Prove that if A is closed and bounded then A is sequentially compact
   • (b) Attempt any Two from the following: 12
   • (i) Prove that a subset K in a discrete metric space (X,d) is compact if and only if K is Paper Subject Code: 24291 Mathematics: Topology of Metric Spaces (R 2022)
   • (ii) Suppose (X,d) is a metric space and C is a non-empty, finite collection of compact subsets of X then show that (J K is a compact subset of X
   • (iii) Prove or disprove :
   • (I) A closed ball in a metric space is compact (II) If A, B be compact subsets of (d being usual), then the set B is also
4. Q4 Attempt any Three from the following: 15 marks
   • (a) Let be metrics on X. Define d: X x X —> Ras d(z,y) = max Show that d is a metric on X
   • (b) Let (X,d) be a metric space and A be any non empty subset of X. Define diameter of set
   • A. Further, find diameter of the following sets in R with usual metric:
   • (c) Show that the equation = x has atleast one solution in R
   • (d) Prove that in a discrete metric space every Cauchy sequence is eventually constant. Hence deduce that a discrete metric space is complete
   • (e) If A, B are compact subsets of R with respect to usual distance, show that A x B isa compact subset of R? with Euclidean metric
   • (f) Determine which of the following subsets of (R*,d), where d is Euclidean distance is com pact. Justify your answer
   • (ii) D <1}

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