(4.2) Let X = {1,2, 3, 4} and 7 = {0, {1,2,3} , {2, 3},{1},X} be the topology defined on X. (a) Show that the only clopen sets in 7 are X and 0, (b) What can you deduce from the results in (4.2a)?

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A: Since you have asked multiple question, we will solve the first question for you. If you want any…

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Q: 13. If t={X,Ø,{2,4},{1,4},{1,2,4},{4}} is a topology on X={1,2,3,4}, let A={1,3,4}, then Int(A)= *…

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Q: Q3. Let N = {1, 2, 3, ...} equipped with the cofinite topology. Prove that M = {7,14, 21} is a…

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Q: Let X = {1, 2, 3, 4, 5, 6} and let T1 = {X, %3D 4, {2}, {1,2}, {2,3,4}, {1, 2,3,4}} and T2 = {X, þ,…

A: If T is the topology on Set X then T contains all subsets of X.

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Q: 4.2 Let Fj, be a closed set for k = 1,2, ... ,n in (S, d). Show that U Fk is closed.

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Q: Prove that l', 12 and lº norms satisfy the properties (N1)-(N3). (N1) (Positivity)||x|| = 0, and ||x…

A: Since you have asked multiple question, we will solve the first question for you. If you want any…