BSc Mathematics SEM V 2019 20 Oct 2019-20 MATHS MATHEMATICS TOPOLOGY OF METRIC SPACES 14.10.19 Question Paper - Mumbai University | munotes

Questions asked in this paper

• (2) Figures to the right indicate marks

1. Q1 Choose correct alternative in each of the following: 20 marks
   • (i) Let A= R: < 1/2}, with usual metric on R, which of the following statements
   • (a) A is an open subset of R
   • (b) A is a closed subset of R
   • (c) A is an open as well as closed subset of R
   • (d) None of these
   • (ii) Which of the following subset of usual metric space R is not. dense?
   • (iii) Let (R,d) be a metric space where d is a discrete metric. Then, which of the following subset of d) is infinite?
   • (a) B(O,0.5) (b) 1) (c) (d) None of these
   • (iv) Which of the following sequences in dis a usual metric from R, is convergent in Q?
   • (v) Every Cauchy sequence is eventually constant in
   • (a) (N,d) where d is usual. (c) (R \ Q,d) where d is usual
   • (b) where dis usual. (d) None of the above
   • (vi) Let d, and be metrics on X such that < di(x,y) < y) for all X where > 0 are constants. Then the statement which is not true is
   • (a) is Cauchy in (X, if and only if is Cauchy in (X,
   • (b) —> pin if and only if x7, — p in (X,
   • (c) is bounded in (X, d,) if and only if (x,) is bounded in (X,
   • (d) None of the above
   • (vii) In R with respect to usual distance is a singleton set when
   • (viii) Which of the following subset of R or R? is compact with respect to the Euclidean metric? Paper Subject Code: 24246 Mathematics: Topology of Metric Spaces
   • (ix) Let A be a compact subset of R. Then
   • (a) A may not be compact. (b) may not be compact
   • (c) 0A may not be compact. (d) None of the above
   • (x) Let (X,d) be a metric space and be a sequence in X such that — as n >
   • (a) :n N} is a compact subset of X
   • (b) N}U {2} is a compact subset of X
   • (c) {xo} is compact only if is a sequence of distinct points
   • (d) None of the above
2. Q2 (a) Attempt any One of the following: 8 marks
   • (i) Every infinite bounded subset of R has a limit point. (distance being usual)
   • (ii) Let (X,d) be a metric space. Prove the following:
   • (1) Arbitrary union of open sets is open (II) A subset G of X is open if and only if it is an union of open balls
   • (b) Attempt any Two of the following: 12
   • (i) Let A be a subset of a metric space (X,d) . Prove that
   • (ii) Let (X,d) be a metric space. X x X — R metric defined as di(x,y) =
3. Q10 Show that d and are equivalent metrics on X
   • (iii) Let (X,|| be a normed linear space and A 4 C X. Show that if U CX is an open set then U + A is open
   • (iv) Show that in (C[0, 1}, || is open in 1], |] where || f = =
4. Q3 (a) Attempt any One of the following: 8 marks
   • (i) Let (X,d) be a metric space be a subset of X. Show that p X is a limit point of and only if there is a sequence of distinct points in A converging to p
   • (ii) State and prove the Nested interval theorem in R
   • (b) Attempt any Two of the following: 12
   • (i) If and (y,) are sequences in a metric space such that 7, —> — q then show that the sequence of real numbers Yn) —> in (R, usual)
   • (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 if and only if G where V is an open set Paper Subject Code: 24246 Mathematics: Topology of Metric Spaces
   • (iii) Check if Cantor’s Theorem is applicable in the following examples. Also, find in each case, where is a sequence of subsets of X CR
   • (i) X and distance d is the usual distance, F, = [—+, 4]
   • (iv) Show that (N,d) is a complete metric space where for m,n N,
5. Q4 (a) Attempt any One of the following: 8 marks
   • (i) Let A be a non-empty subset of the metric space (R,d) where d is the usual metric Prove that A is sequentially and only if A satisfies the Bolzano-Weierstrass
   • (ii) Show that a compact subset of a metric space is closed and bounded. Give an example to show that a closed and bounded subset. need not. be compact
   • (b) Attempt any Two of the following: 12
   • (i) Suppose (X,d) is a metric space and C is a non-empty collection of compact subsets of X then show that if C is finite then K is a compact subset of X
   • (ii) Show that = R? y? is a compact subset of R?, distance being
   • (iii) If X = [0, 1] (R,d), where d is the discrete metric, show that the open cover { of X has no finite subcover
   • (iv) Show that R": < (n+1)?} is a compact subset of being Euclidean
6. Q5 Attempt any Four of the following: 20 marks
   • (a) State and prove Hausdorff property in a metric space
   • (b) Show that S = < x? <5} is both open and closed in the subspace Q of R with
   • (c) f [a,b] R is a continuous such that f takes only rational values then show that f is
   • (d) Prove or disprove : If and are equivalent metrics on X and is a complete metric space then (X,d2) is also a complete metric space
   • (e) If A,B are compact subsets of IR with respect to usual distance, show that A x B isa compact subset of R? with Euclidean metric Consider the set A = [0,1] in the metric space (R,d),d being the discrete metric. Show that the open cover of A has no finite subcover

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