BSc Mathematics SEM V ATKT May 2016 ATKT MATHEMATICS SEMV 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) Show that in a metric space (X, d)
   • (1) an arbitrary union of open sets is an open set (II) a finite intersection of open sets is an open set
   • (ii) Define an open ball B(z,r) 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
   • (b) Attempt any Two from the following: 12
   • (i) Define a normed linear space (X, || ||). Show that in a normed linear space ||),
   • (ii) Define distance of a point p from a set A in a metric space If A then show
   • (iii) are two metrics on R? defined by where x = (21, = (y1, y2) R? Show that and dz are equivalent metrics
   • (iv) State and prove the Hausdorff property metric space (X,d)
2. Q2 (a) Attempt any One from the following: 8 marks
   • (i) Show that for a subset of a metric space (X,d), closed if and only if contains all its limit points
   • (ii) (X,d) isa metric space and A,B C X. Then prove the following:
   • (1) and equality may not hold
   • (b) Attempt any Two from the following: 12
   • (i) Let (X,d) be a metric space. If sequence is a Cauchy sequence in (X,d) and the sequence has a convergent subsequence which converges to p X, then prove that the sequence (z,,) also converges to p
   • (ii) Show that for any subset A of a metric space (X,d), 6(A) = 6(A) where 6A indicates the diameter of A
   • (iii) Let and dy be metrics on a non-empty set X such that there exists > 0 such that. kidi(x,y) < do(a,y) < kedi(x,y) Vx,y X then show that a sequence (z,,) is bounded in (X,d,) if and only if is bounded in (X,
   • (iv) Which of the following are dense subsets of R with the usual distance? Justify your
   • (1) Q (II) Z (III) R\Z Paper Subject Code: 24142 Mathematics: Topology of Metric Spaces. (Rev.) (R-2016)
3. Q3 (a) Attempt any One from the following: 8 marks
   • (i) Show that if a subset K of R” is sequentially compact then it is closed and bounded
   • (ii) Show that a subset A of R” has the Bolzano-Weierstrass property if and only if
   • (b) Attempt any two from the following: 12
   • (i) If A, B are compact subsets of R then show that A+ B is also a compact subset of R
   • (ii) Prove that a subset of a discrete metric space is and only if it is finite
   • (iii) Show that a closed subset of a compact metric space is compact
   • (iv) Consider the metric space where dis the usual distance. Show that N} is an open cover of (0, 1). Is (0,1) a compact set? Justify your answer
4. Q4 Attempt any Three from the following: 15 marks
   • (a) Prove or disprove: Every open ball in (N,d,) is an open ball in (N,d) where is the discrete metric on N and d is the usual metric
   • (b) Show that U = {(z,y) R? 2x — 3y is an open subset of R? with the Euclidean
   • (c) Consider the sequence (f,) of functions in 1] defined by Show that {f,} is Cauchy w.r.t. |] ||; where = | f(t)|dt
   • (d) If and are sequences in a metric space (X,d) such that —> p and y, — q then show that the sequence of real numbers Yn) —> in (R, usual)
   • (e) Prove or disprove:
   • (1) Interior of a compact set is compact (II) Every compact subset of (R, usual) has a limit point
   • (f) (X,d) be a compact metric space. If {A,,} is a sequence of non-empty closed sets in X such that C A, for each n N then show that An #9

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