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I'm not sure about my answers, any help is highly appreciated.
Let (N, U) be a topological space, where N is the set of natural numbers (without 0), and U = {0} U {Oi, i is from N}, where Oi = {i, i+1, i+2, ...} and {0} is the empty set. One has to find the interior (Int) and closure (Cl) of these sets:
(a) A = {n from N : (n - 7)/(n - 11) > 0}
(b) A = {13, 5, 2010}
So, for (a):
Obviously, A = {n from N : n > 11 or n < 7} = N \ {7, 8, ... , 11}. The interior of A, Int(A), is by definition the union of all open subsets contained in A, and the open subsets in the topology (N, U) are elements of the family U or their unions and finite intersections. So Int(A) = {0} (the empty set, since no open subset in U can contain A. Now, the closure of A, Cl(A), is by definition the intersection of all closed subsets which contain A, i.e. the smallest one of them. So, Cl(A) = N, and N is closed, since its complement is the empty set, which is open. (I feel I'm missing something huge here.)
(b) Again, Int(A) = {0}. What would Cl(A) be? The first guess is Cl(A) = {5, 6, ...}, but is this set closed? Its complement is {1, 2, 3, 4}, but this is no open set contained in U?
I hope I didn't cause much confusion, but I need to solve this problem in order to clear out my way of thinking.
Let (N, U) be a topological space, where N is the set of natural numbers (without 0), and U = {0} U {Oi, i is from N}, where Oi = {i, i+1, i+2, ...} and {0} is the empty set. One has to find the interior (Int) and closure (Cl) of these sets:
(a) A = {n from N : (n - 7)/(n - 11) > 0}
(b) A = {13, 5, 2010}
So, for (a):
Obviously, A = {n from N : n > 11 or n < 7} = N \ {7, 8, ... , 11}. The interior of A, Int(A), is by definition the union of all open subsets contained in A, and the open subsets in the topology (N, U) are elements of the family U or their unions and finite intersections. So Int(A) = {0} (the empty set, since no open subset in U can contain A. Now, the closure of A, Cl(A), is by definition the intersection of all closed subsets which contain A, i.e. the smallest one of them. So, Cl(A) = N, and N is closed, since its complement is the empty set, which is open. (I feel I'm missing something huge here.)
(b) Again, Int(A) = {0}. What would Cl(A) be? The first guess is Cl(A) = {5, 6, ...}, but is this set closed? Its complement is {1, 2, 3, 4}, but this is no open set contained in U?
I hope I didn't cause much confusion, but I need to solve this problem in order to clear out my way of thinking.