BSc Mathematics SEM VI 2016 17 2016-17 Topology Question Paper - Mumbai University | munotes

Questions asked in this paper

1. Q2 (a) Attempt any One of. the following: 8 marks
   • (i) Show that for a subset F of a (X,d), the following statements are equiv
   • (I) F is closed (II) F contains all its limit
   • (ii) Let 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 a sequence of distinct points in A converging
   • (b) Attempt any Two of the following: f 12
   • (i) Let (distance being usual), where A =Nand B= \n + A be a subset of a metric space (X,d) . Prove that (II) (X \ A)? = Show that S = {(z,y) R? = 1} is a closed subset of , where the distance
   • (iv) Prove that a subset A of a metric space is dense in X if and only if GNA for each non-empty open subset G of X
2. Q3 (a) Attempt any One of the following | act then prove that has Bow
   • (i) If K is such that roperty Euclidean, ic space is not
   • (ii) Show that a compact subset 0 bounded gubset of metric Give an example to show that a|close
   • (b) Attempt any two: and only if it 1s finite
   • (i) Prove that a subset of a discrete space is compact converges to some
   • (ii) (X,d) is a metric space and (Zn) is a x such by .using then show that comp definition of compactness. i
   • (iii) Let A, B be compact subsets.of d), distance d being usual. Show the compact subset of d’) where id’ is the Euclidean distance
   • (iv) Consider the metric space (IR,d), where d is the usual distance Show
3. Q1 :n is an open (0,1). Is (0,1) compact ? Justify your
4. Q4 Attempt any Three of the following: 15 marks
   • (a) Prove or disprove : If (X,d) metric space and X,r,s > B(z,r) = 8);
   • (b) Show that || || isa norm on X, where X = and = max 1 2} for A= (ai;)
   • (c) Let (X,d) be a discrete metric space and AC X. Then prove that = A
   • (d) Consider the sequence of functions in 1} defined by Show that {f,} is Cauchy w.r.t. || where = i | f Let A= {(z,y) R? 1}. Determine whether A is compact. Justify your answer
   • (f) Prove or disprove : | A closed ball B [z, r] in a metric space is compact

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