Question 1 a) Let (X, T) be a topological space and Y = XU {a}, where...

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Question 1 a) Let (X, T) be a topological space and Y = XU {a}, where a ¢ X. Prove that Ta = {{a}U0:O E T}U {0} is a topology on Y. b) Let X = {a, b, c, d}. Let T be a topology on X given by T = {0, {b}, {d}, {a, b}, {b, d}, {b, c}, {a, b, d}, {a, b, c}, {b, c, d}, X}. Explain why (X,T) is not Hausdorff. c) Find three open sets in the real line that have the same boundary. Is it possible to increase the number of such sets? [5,3,6] Question 2 Let X be a topological space. a) Prove that À = Á for each subset A of X. b) If ACY and Y is a subspace of X. Let TA= Trace topology of A in X and (TA)Y= Trace topology of A in Y. Prove that TA = (TA)Y. c) Show that (FUÁ Á) = (FUA)° for every closed subset F of X and every ACX. Question 1 a) Let (X, T) be a topological space and Y = XU {a}, where a ¢ X. Prove that Ta = {{a}U0:O E T}U {0} is a topology on Y. b) Let X = {a, b, c, d}. Let T be a topology on X given by T = {0, {b}, {d}, {a, b}, {b, d}, {b, c}, {a, b, d}, {a, b, c}, {b, c, d}, X}. Explain why (X,T) is not Hausdorff. c) Find three open sets in the real line that have the same boundary. Is it possible to increase the number of such sets? [5,3,6] Question 2 Let X be a topological space. a) Prove that À = Á for each subset A of X. b) If ACY and Y is a subspace of X. Let TA= Trace topology of A in X and (TA)Y= Trace topology of A in Y. Prove that TA = (TA)Y. c) Show that (FUÁ Á) = (FUA)° for every closed subset F of X and every ACX.

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