BSc Mathematics SEM VI ATKT 2018-19 ATKT Maths III Matric Topology 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) State and prove Cantor’s Intersection Theorem for a metric space (X, d)
   • (ii) Let f : [a,b] —> R be continuous. If f(a) and f(b) have opposite signs then using Nested Intervals Theorem, prove that there exists c (a,b) such that f(c)
   • (b) Attempt any Two from the following: 12
   • (i) Let (X,d) be a complete metric space and (Y,dy-) is a subspace of (X,d). If (Y,dy) is complete then show that Y is a closed subset of X
   • (ii) Prove that a finite metric space is complete
   • (iii) Show that [0, 1] is uncountable
   • (iv) Prove that the set of real numbers R is complete with respect to the usual distance
2. Q2 (a) Attempt any One from the following: 8 marks
   • (i) Let f : (X,d) — be a function. Show that f is continuous at p X if and only if for each sequence (x,,) in X converging to p, the sequence converges to
   • (ii) Let (X,d) and be metric spaces. is compact and X —> Y isa continuous function, then show that f(X) is subset of Y
   • (b) Attempt any Two from the following: 12
   • (i) Let (X,d) and (Y,d) be metric spaces then show that f : X —> Y is continuous if and only if , for each subset B of Y
   • (ii) Let (X,d) and be metric spaces and —> Y be continuous on X. Show that X : f(x) = g(x)} is a closed subset of X
   • (iii) Show that the identity function 7: (R,d) —> R is discontinuous everywhere in IR where d is the usual distance and is the discrete metric
   • (iv) Let (X,d) and (Y,d') be metric spaces and D C X be a dense subset of X. If f : X Y continuous onto map, show that f(D) is dense in Y
3. Q3 (a) Attempt any One from the following: 8 marks
   • (i) Prove that.a subset of R is connected if and only if it is an interval. (Distance in R
   • (ii) Prove that a metric space is connected if and only if every continuous function from X to is a constant function
   • (b) Attempt any Two from the following: 12
   • (i) If (X,d) is a metric space and A, B are conncected subsets of X such that AN BZ 0 then prove that AUB is a connected set
   • (ii) Prove that a convex subset of a normed linear space is path connected
   • Q. P. Code : 51411
   • (iii) Prove or disprove: The subset R? y 0} of (R*,d) (d being Euclidean distance ) is connected
   • (iv) If (X,d) be a connected metric space and f : X —> Z (distance in Z being usual distance) is a continuous function then prove that f is a constant function
4. Q4 Attempt any Three from the following: 15 marks
   • (a) Use the intermediate value property to show that there is a square whose diagonal has length between r and 2r and has area equal to half the area of the circle of radius r
   • (b) Check if Cantor’s Theorem is applicable in the following examples. Also , find F, in each case, where (F;,) is a sequence of subsets of Rand the distance in R is usual (II) = (0, +)
   • (c) If T: 0, 3| defined as T(x) then show that T is a contraction map on . Does T have any fixed points? If yes, how many? Justify your answer
   • (d) Discuss the uniform continuity of f : —+ R (distance being usual), defined by
   • (e) Prove or disprove: If and are connected then A is connected
   • (f) Show that E = {(x,y) 2 > — y? = 1} is path connected

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