BSc Mathematics SEM V 2018 19 2018-19 Maths 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) Let (X,d) be a metric space and A C X. Show that
   • (1) is an open set and is the largest open in A (II) A is open if and only if A=
   • (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 converse need not be true
   • (b) Attempt any Two of the following: 12
   • (i) Let ||) be a normed linear space and X. Show that is an open set then U + A is open
   • (ii) Define distance of a point p from set A in a metric space (X,d). If X then show
   • (iii) 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
   • (iv) Let dz be metrics on X. Defined : X xX —> = max y), y)} Show that d is a metric on X
2. Q2 (a) Attempt any One of the following: 8 marks
   • (i) 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
   • (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 = VOY where V is an open set
   • (b) Attempt any Two of the following: 12
   • (i) Let (X,d) be a metric space C X. If G C X is an open set such that GN A = 0 then show that GN A =
   • (ii) Let and be metrics on a non-empty set X such that there exists > 0 such that < X then show that (z,,) is bounded in if and only if is bounded in (X,
   • (iii) Let A be a subset of a metric space (X,d) . Prove that Paper Subject Code: 24142 Mathematics: Topology of Metric Spaces. (Rev.) (R-2016)
   • (iv) Let d and d, be equivalent metrics on X. If pin (X,d) then prove that
3. Q3 (a) Attempt any One of the following: 8 marks
   • (i) Show that if a subset K of R” is sequentially compact then it is closed and bounded
   • (ii) If J = [ay, bi] x be] x X C R” then prove that J is compact (distance in
   • (b) Attempt any Two of the following: 12
   • (i) If A, B are compact subsets of then show that A+ B is also a compact subset of
   • (ii) Show that closed subset of compact metric space is compact
   • (iii) Show that 1], || where = sup 4 [0,1] is not compact by considering the open cover N} of
   • (iv) Prove or disprove:
   • (1) Interior of a compact set is compact (II) Closure of a compact set is compact
4. Q4 Attempt any Three of the following: 15 marks
   • (a) State and prove Hausdorff property in a metric space (X,d)
   • (b) Show that dis a
   • (c) Which of the following are dense subsets of R with usual distance? Justify your answer
   • (I) Z R\O (III) R\Z
   • (d) Let (X,d) bea metric sapce (X,d). Show that every Convergent sequence X is Cauchy
   • (e) Give an example of two metrics and dz on X such that is compact but (X, d2)
   • (f) Which of the following subsets of d), (d being Euclidean distance) are compact? Justify
   • (ii) B=

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