BSc Mathematics SEM VI 2018 19 May 2018-19 MATHEMATICS TOPOLOGY OF METRIC SPACES & REAL ANALYSIS Question Paper - Mumbai University | munotes

Questions asked in this paper

• (2) Figures to the right indicate marks

1. Q1 Choose the correct alternative for each of the following: 20 marks
   • (i) If f : (R,d) —> (R,d), with d as the usual distance, is a continuous function then
   • (a) a closed subset of R. (c) an open subset of R
   • (b) a bounded subset of R. (d) none of the above is true
   • (ii) Let (X,d) be a discrete metric space and be any metric space. If X —> Y, then
   • (a) an uniformly continuous function on X
   • (b) a bounded function on X
   • (c) continuous but not uniformly continuous function on X
   • (d) none of the above is true
   • (iii) Consider the metrics d and d; on N, where dis the induced distance from R with the usual metric and = for m,n N. Let i: N—> N denote the identity map on
   • N. Then
   • (a) (N,d) —> (N, is continuous but 7: —> (N,d) is not continuous
   • (b) 7: (N,d) —> is not continuous
   • (c) d) is not continuous
   • (d) None of the above
   • (iv) In (R?,d) where d is the Euclidean distance, the following set is not connected
   • (b) R? \ {(0,0)} (d) None of the above
   • (v) Let A and B be connected subsets in a metric space (X,d) and AC C C B Then,
   • (a) C is connected. (c) Cis connected
   • (b) is connected. (d) CNA is connected
   • (vi) In R? with the Euclidean metric, which of the following sets is convex?
   • (b) {(x,y) xy = 0} (d) None of these
   • (vii) Let (X,d) bea connected metric space and f : X Z be a continuous map. Then, f is
   • (a) an onto function. (c) a bijective function Paper Subject Code: 88639 Mathematics: Topology of Metric Spaces & Real Analysis
   • (viii) Let = for x R. and g,(x) = Va R. Then,
   • (a) {fr} and {g,} are uniformly convergent on R
   • (b) {fn} and {g,} are not pointwise convergent on R
   • (c) {gn} is uniformly convergent on R but is not
   • (d) {fn} is uniformly convergent on R but {g,} is not
   • x) Th
   • (ix) The series
   • (a) uniformly convergent on R
   • (b) not uniformly convergent on [—a,a] where 0<a< 1
   • (c) uniformly convergent on where <
   • (d) none of the above
   • (x) If R is the radius of convergence of the power series then the radius of convergence
2. Q2 (a) Attempt any One of the following: 8 marks
   • (i) Let (X,d) and be metric spaces. If (X,d) is compact and f : X —> Y isa continuous function, then show that f(X) is a compact subset of Y
   • (ii) Let f : (X,d) be a function. Prove that. f is continuous on X if and only if for each open subset G of Y, is an open subset of X
   • (b) Attempt any Two of the following: 12
   • (i) Let (X,d) be metric spaces then show that f : X Y is continuous if and only if , for each subset B of Y
   • (ii) (X,d) isa metric space and f : (X,d) —> (X, d) isa function such that f(y)) < d(x,y) whenever 7 # y. Let S={x eX: f(x) =x}. Prove that
   • (1) f is continuous on X (II) has at most one element
   • (iii) Let (X,d) and be metric spaces. When is f : X —> Y said to be uniformly continuous? Show that = is uniformly continuous on R (under the
   • (iv) Let and be equivalent metrics on X and (Y,d) be any metric space. If f : (X,d,) (Y,d) and g : (Y,d) —> (X,d,) are continuous maps on X and Y re spectively, then prove that f : (X,d2) —> (Y,d) and g: (Y,d) —> are also Paper Subject Code: 88639 Mathematics: Topology of Metric Spaces & Real Analysis
3. Q3 (a) Attempt any One of the following: 8 marks
   • (i) Define a connected metric space and 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
   • (ii) Prove that a metric space is connected if and only if every continuous function from X to {1,—1} is a constant function
   • (b) Attempt any Two of the following: 12
   • (i) Let (X,d) be a metric space such that for X there exists a connected subset A of X such that x,y A. Prove that X is connected
   • (ii) Prove that in a normed linear space, an open ball is a convex set
   • (iii) Prove that if a subset of R is connected then it is an interval. (Distance in R being
   • (iv) Let A= {(z,y) R? and B= {(z,y) R? =1} in where d is the Euclidean metric. Show is a path connected set
4. Q4 (a) Attempt any One of the following: 8 marks
   • (i) Let {fn} be a sequence of real valued functions defined on a set S C R such that fn — f uniformly on S. If each f, is bounded then prove the following:
   • (1) f is bounded (II) There exists a R* such that for all n N and for all x S
   • (ii) State and prove the Cauchy criterion for uniform convergence of a series of functions
   • (b) Attempt any Two of the following: 12
   • (i) Find the pointwise limit of the sequence of functions : R, ; Ts the pointwise limit bounded?
   • (ii) Find the radius of convergence and interval of convergence of the following power
   • (iii) Show that the series of functions converges uniformly on > 0
   • (iv) Consider the power series cnx” with integer coefficients. If c, 0 for infinitely many n, then show that its radius of convergence is at most 1
5. Q5 Attempt any Four of the following: 20 marks
   • (a) Prove or disprove: Continuous image of an open ball is an open ball Paper Subject Code: 88639 Mathematics: Topology of Metric Spaces & Real Analysis
   • (b) Let f : [a,b] [a,b] be continuous on and differentiable on (a,b). R with that |f’(x)| < c, Vx (a,b) then prove that f is a contraction of [a,
   • (c) Let (X,d) be a connected metric space. If f : X —> where is a discrete metric, is a continuous function then prove that f is a constant function
   • (d) Prove that there does not exist a continuous onto function from the set A = R?
   • (e) If < oo, then prove that the series a, cosnx and a, sinnx converge on R
   • (f) Let R, f,(x) = + 4. Given that f, —> f uniformly on 1] where f(a) = |x| for Find tim

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