BSc Mathematics SEM V 2016 17 2016-17 Mathematics Topology Of Matric 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) In a metric space (X,d), prove that arbitrary union of open sets is open in X. Give an example to show that arbitrary intersection of open sets is not open in X
   • (ii) Let (X,d) be a metric space and A, B Show that (II) (AN = (III) C and the inequality may be strict
   • (b) Attempt any Two of the following: 12
   • (i) Prove or disprove: Let be equivalent metrics on a non-empty set X. If (x,) is bounded in then is bounded in (X,
   • (ii) (Z,d) and (Z, d,) are metric spaces where d is the usual metric (induced from R ) and d, is the discrete metric in Z. Prove that d are equivalent metrics
   • (iii) Let be metrics on X. Defined: XxX —> = max y), do(z, y)} Show that d is a metric on X
   • (iv) || ||, and |] are norms on R? where for = = = x7. Show < and < V2 for R?
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). Fis closed (II) contains all its limit points
   • (ii) Let (X,d) be a metric space and A be a subset of X. Show that p X is a limit point of A if and only if there is of distinct points in A converging to p
   • (b) Attempt any Two of the following: 12
   • (i) Let A, B C R (distance being usual), where A = N and B = i" + N,n>
   • (ii) Let A be a subset of a metric space (X,d) . Prove that
   • (iii) Show that = 1} is a closed subset of , where the distance
   • (iv) Prove that a subset A of a metric space (X,d) is dense in X if and only if for each non-empty open subset G of X
   • Q. P. Code: 05723
3. Q3 (a) Attempt any One of the following: 8 marks
   • (i) If kK C R” is such that K is compact then prove that AK has Bolzano-Weierstrass
   • (ii) Show that a compact subset of where d Euclidean, is closed and bounded Give an example to show that a closed and bounded subset of a metric space is not
   • (b) Attempt any two: 12
   • (i) Prove that a subset of a discrete metric space is compact if and only if it is finite
   • (ii) (X,d) is a metric space and is a sequence in X such that converges to some point p X. If S = :n N}U {p} then show that S is compact by using the definition of compactness
   • (iii) Let A,B be compact subsets of d), distance d being usual. Show that A x B isa compact subset of where d’ is the Euclidean distance:
   • (iv) Consider the metric space (R,d), where dis the usual distance . Show that {(4, 1) :n N} is an open cover of (0,1). Is (0,1) Justify your answer
4. Q4 Attempt any Three of the following: 15 marks
   • (a) Prove or disprove : If (X,d) metric space and x,y X,r,s > = then either = y orr =s
   • (b) Show that || || on X, where X = M2(R) and Al] = max <2} for A= (a;)
   • (c) Let (X,d) bea discrete metric space and A C X.'Then prove that = A
   • (d) Consider the sequence of functions in 1] defined by Show that {f,} is Cauchy w.r.t. || ||; where = Let A= R? <1}. Determine whether A is compact. Justify your answer
   • (f) Prove or disprove : A closed ball in a metric space is compact

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