BSc Mathematics SEM V 2016 17 2016-17 Mathematics VI 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) Let (X,d) be a metric space. Define limit point of F Also show that F’ is closed if and only if contains all it’s limit points
   • (ii) 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
   • (b) Attempt any Two questions: 12
   • (i) Show that U = {(z,y) 2x +3y < 1} is an open subset of R? with Euclidean
   • (ii) Let (X,d) be a metric space and X. Show that is an open set and is the largest open set contained in A
   • (iii) Prove that in any metric space (X,d), A is closed if and only if 0A C A where OA denotes the boundary of A
   • (iv) State and prove Hausdorff property in a metric space (X,d)
2. Q2 (a) Attempt any one question: 8 marks
   • (i) If in a metric space (X,d), for every decreasing sequence of non-empty closed sets with 0, we have is a singleton set then prove that (X,d) is
   • (ii) Let (X,d) be a metric space and A C X. Prove that p A if and only if there is a sequence of points in A converging to p
   • (b) Attempt any Two questions: 12
   • (i) Prove or disprove: Let be equivalent metrics on a non-empty set X. If (x,) is bounded in (X, d;) then is bounded in (X, d2)
   • (ii) Check if Cantors Theorem is applicable in the following examples. Also , find in each case, where is a sequence of subsets of R and the distance in R is usual (II) = (0, +)
   • (iii) Prove that in a discrete metric space every Cauchy sequence is eventually constant Hence deduce that a discrete metric space is complete
   • (iv) Show that a sequence (a,,) in (R?,d) (where d is Euclidean distance) converges to a point p = p2) R? if and only if —> p; for 1 <i < 2, in R with respect to the usual distance, where =
   • Q. P. Code: 05719
3. Q3 (a) Attempt any One from the following: 8 marks
   • (i) Let f : (X,d) — (Y,d’) be a function. Show that f is continuous at p X if and only if for each sequence in X converging to p, the sequence converges to
   • (ii) Let (X,d) and be metric spaces. Show that —+ Y is continuous on X if and only if for each subset A of X, f(A) C (f(A))
   • (b) Attempt any Two questions: 12
   • (i) If f,g : (X,d) — p) are continuous on X and f(x) = g(x) Vr A, AC X , then show that = g(x) Vx A
   • (ii) Prove or disprove: Continuous image of of an open open
   • (iii) Let (X,d) and (Y,d’) be metric When is X —> Y said to be uniformly continuous? Show that f(x) = is uniformly continuous on R (under usual
   • (iv) Prove every function —> (Y,d) where (Y,d) is any metric space is
4. Q4 Attempt any Three questions: 15 marks
   • (a) Show that U = {(z,y) R? 2x+3y is an open subset of R? with Euclidean metric
   • (b) Show that d: N x N —> R is a metric on N where d is defined as follows:
   • (c) Let X = and d, be the metric induced by || ||; on X. = | f(t)|dt). Show that the following sequence of functions {f,,} is bounded in (X,
   • (d) Let (X,d) and be metric spaces. Show that if f : X —> Y is uniformly continuous on X and if in X is Cauchy then show that the sequence is Cauchy in Y Let be a metric space and let A C X, If X —> R is defined by = d(z, A) Then show that d4 is continuous on X. (distance in R being usual)
   • f) Prove or disprove: Continuous image of of an open set is open

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