Let (X, T) be any topological space and A any subset of X. The largest open...

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Let (X, T) be any topological space and A any subset of X. The largest open set contained in A is called the interior of A and is denoted by Int(A). [It is the union of all open sets in X which lie wholly in A.] (1) Prove that in R, Int([0, 1]) = (0, 1). (ii) Prove that in R, Int((3,4)) = (3,4). (iii) Show that if A is open in (X, T) then Int(A) = A. (iv) Verify that in R, Int({3})=0. (v) Show that if (X, T) is an indiscrete space then, for all proper subsets A of X. Int(A) = Ø. (vi) Show that for every countable subset A of R. Int(A) = Ø. Let (X, T) be any topological space and A any subset of X. The largest open set contained in A is called the interior of A and is denoted by Int(A). [It is the union of all open sets in X which lie wholly in A.] (1) Prove that in R, Int([0, 1]) = (0, 1). (ii) Prove that in R, Int((3,4)) = (3,4). (iii) Show that if A is open in (X, T) then Int(A) = A. (iv) Verify that in R, Int({3})=0. (v) Show that if (X, T) is an indiscrete space then, for all proper subsets A of X. Int(A) = Ø. (vi) Show that for every countable subset A of R. Int(A) = Ø.

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let xd be any topological space and A be any subset of x The set cont ained 18 Jat A lie wholly in 4 ... View the full answer