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Topology Homework 3

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Topology Homework 3 Section 3 1 Section 3 3 Samuel Otten 3 1 1 Proposition The intersection of finitely many open sets is open and the union of finitely many closed sets is closed Proof Note that S1 S2 S3 Sn S1 S2 S3 Sn for any family of sets Si i N and any natural number n Thus for an intersection of finitely many open sets we can take the intersection pairwise and by Definition 1 ii each set along the way is open and the end result is open Similarly S1 S2 S3 Sn S1 S2 S3 Sn Thus for a union of finitely many closed sets we can take the union pairwise and by Proposition 1 b each set along the way is closed and the end result is closed 6 Proposition Let F N be closed if F contains a finite number of positive integers or F N Then the set F of all closed subsets of N thus defined forms a topology on N Moreover this topology is not induced by any metric Proof We shall employ Proposition 2 of the text to demonstrate that N F is a topological space First by definition F N is closed and is closed because it contains a finite number of positive integers namely zero Second let F1 and F2 be elements of F If at least one of F1 and F2 is equal to all of N then F1 F2 N which is closed If this is not the case then both contain a finite number of positive integers Thus their union contains a finite number of positive integers and is closed by definition Third we consider the intersection of a family of elements of F The intersection of sets containing finitely many elements must itself contain finitely many elements The worst that the intersection could do is result in N itself but this is closed Therefore F defines a topology on N Suppose a metric D on N did induce N F Since N D is a metric space it must satisfy Proposition 12 of Chapter 2 which says that for any closed subset of the space and any point in its complement there exist disjoint open subsets such that the closed subset is contained in one and the point is contained in the other Consider F 1 2 3 and x 4 It is obviously

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