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Unformatted text preview: 147 HW1 Solutions 2.13.1) Let X be a topological space; let A be a subset of X . Suppose that for each x ∈ A there is an open set U containing x such that U ⊂ A . Show that A is open in X . Solution: Let x ∈ U x ⊂ A for each x ∈ A such that U x are all open. We know these sets, U x , exist by the problem statement. Then S x ∈ A U x = A , so A is open. 2.13.2) Consider the nine topologies on the set X = a,b,c indicated in Example one of § 12. Compare them; that is, for each pair of topologies, determine whether they are comparable, and if so, which is finer. Solution: 1 is coarser than all others. 2 is finer 1. 3 is finer than 1. 4 is finer than 1,3. 5 is finer than 1. 6 is finer than 1,2,3,4. 7 is finer than 1,2,3. 8 is finer than 1,2,3,7. 9 is finer than all others. 1 2.13.4) (a) If {T α } is a family of topologies on X , show that T T α is a topology on X . Is S T α a topology on X ?...
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This note was uploaded on 05/19/2010 for the course MAT mat147 taught by Professor Derekwise during the Spring '10 term at UC Davis.
 Spring '10
 derekwise

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