BSc Sem VI 2022 2023 Apr 2023 MATHEMATICS TOPOLOGY OF METRIC SPACES AND REAL ANALYSIS 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) Let f : (X,d) — (Y,d’) be a function. Show that f is continuous at p X if and only if for each sequence (x,,) in X converging to p, the sequence converges to
   • (ii) Let (X,d) and be metric spaces. If (X,d) is a compact metric space and —Y is function, then show that f(X) is a compact subset of Y
   • (b) Attempt ANY TWO from the following: 12
   • (i) Let (X,d) be a metric space and f,g : (X,d) —> R (usual distance) be continuous on
2. Q10 Show that f — R is also continuous on X. Is the converse true?
   • (ii) Let f,g : R —> R be continuous functions (with respect to usual distance). Let (R?,d) —> be defined by h(x, y) = (f(x), g(y)). Show that h is continuous on (R?,d) where d is Euclidean distance
   • (iii) Prove or disprove: If R — R are uniformly continuous on a nonempty set A C R then the product function f -g is uniformly continuous on A
3. Q2 (a) Attempt ANY ONE from the following: 8 marks
   • (i) Define separated sets in a metric space, disconnected metric space. Prove that a metric space (X,d) is disconnected if and only if there exists a nonempty proper subset of X which is both open and closed in X
   • (ii) Define connected subset of a metric space. Prove that if a subset of R is connected then it is an interval. (Distance in R being usual). Is Q connected? Justify
   • (b) Attempt ANY TWO from the following: 12
   • (i) Show that the set S = R?0 <2 < 2,1 < 5} is a convex set in where d is the Euclidean distance
   • (ii) If (X,d) be a connected metric space and f : X —+ Z (distance in Z being usual distance) is a continuous function then prove that f is a constant function
   • (iii) Let (X,d) be a metric space and A, B be conncected subsets of X such that ANB 0 Prove that AU B is a connected set
4. Q3 (a) Attempt ANY ONE from the following: 8 marks
   • (i) Let {f,} be a sequence of real valued R—integrable functions defined on such that f, —> f uniformly on [a,b] . Prove that f is R— integrable on and Paper Subject Code: 88679 Mathematics: Topology of Metric Spaces and Real Analysis(R-2023)
   • (ii) Prove that if the power series converges at x, 0 and diverges at R then the power series converges for all R with |x| < |x,| and diverges for all x R with |x| >
   • (b) Attempt ANY TWO from the following: 12
   • (i) State and prove Cauchy Criterion for Uniform Convergence of a Series of real valued functions defined on a subset S' of R
   • (ii) Show that is uniformly convergent on [0, A], A > 0
   • (iii) Let f, : [-1,1] R with f,(x) = 4. Given that f, —> f uniformly on where f(x) = for x Find lim fr(x) da
5. Q4 Attempt ANY THREE from the following: 15 marks
   • (a) Let (X,d) and be metric spaces. If f : X is uniformly continuous on X and is a Cauchy sequence in X, then show that the sequence is Cauchy in Y
   • (b) Define a contraction map. T : — [0 is defined as T(x) = x”, show that T is a contraction map on
   • (c) Let (X,d) be a metric space. If A is a finite subset of X having more than one element, show that A is disconnected
   • (d) Prove or disprove: The subset y 4 0} of d being Euclidean distance,
   • (e) Find the set of convergence of the power series
   • (f) For each n let f, : —> R be defined by = Show that converges pointwise but not uniformly on [0, 1]

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