(r e Rla co) c R. Which of the following topologies on the set R have 0 as a lim
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(r e Rla co) c R. Which of the following topologies on the set R have 0 as a limit 49. Let A point Discrete. Trivial Circle all that apply: Standard Ra 50. consider the sequence of reals given by am 3 i/n. For which topologies on R does this sequence converge Discrete. Circle all that apply: Standard RI Trivial 51. Consider the sequence of reals given by a 1/n. For which topologies on R does this sequence converge? Discrete. Circle all that apply: Standard Ru Trivial 52. Let x (1.2, 3, 4) and T be the topology on x given by (a, 2). 3, Consider the sequence an 2 (-1) from Which of the following elements does this sequence converge to? Circle all that apply. Circle 1 2 3 4 53. Let X (1,2, 3, 4) and T be the topology on X given by to, x 12.3), 12h, 13), 12.3.4 Consider the sequence an 3 (-1)" from X X. Which of the following elements does this sequence converge to? Circle all that apply. Circle 1 2 3 4 54. Let x- (1,2, 3, 4) and T be the topology on X given by 10,x,12,3),12),13), 12, 3, 4)). Consider the sequence an 3 from X X. Which of the following elements does this sequence converge to? Circle all that apply. Circle 1 2 3 4 55. Let x- (1.2, 3, 4), T (X,0. (1.3), (2,4). Which of the following functions f X X are continuous circle all that apply: f(r) 3 f(z) 5 z f(z) 56. Let x (1, 2, 3, 4), T fx, 0,{1, 2, 3h, 12, 3), 12, 3,4)). Which of the following functions f x x are continuous? Circle that apply: f(r) 3 f(r) 5 r f(r) allExplanation / Answer
64) not connected, since the element 2 is there in all open sets and so the set X cannot be represented as a union of two disjoint subsets.
52) 3 is the only limit point but not 1 since there is no open set that contains 1, where as there is an open set {3} that contains infinitel many elements from the sequence (namely the element 3 which repeats infinitely many times).
53) 2 and 4 are both limit points , same argument as above but here we have open sets {2} and {2,3,4} for 2 and 4 respectively.
54) 3 since its a constant sequence and {3} is an open set.
55) f(x) = 5-x and f(x) = x are continous functions since inverse of all open sets in X are open in this mapping,
56) same thing as above.
57) same as above.
58) all three are continous, since inverse map of the constant function f(x) = 3 is an open set in X.
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