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Unformatted text preview: M 453 S A 2 S 20, 2004 Exercise 1 from Sections 12 & 13, page 83 We will show that A is open by exhibiting it as a union of open sets. For each x ∈ A , let U x be the open set containing x such that U x ⊂ A . It is easy to see that A = S x ∈ A U x , so A is open. Exercise 4 from Sections 12 & 13, page 83 (a) Let T = T T α . To show that T is a topology, we have to verify that T satisfies the three properties in the definition of a topology: (1) Are ∅ and X in T ? Yes, because ∅ and X are in T α for each α . (2) Let { U β } be a collection of open sets in T . Since T is the intersection of the topologies T α , { U β } is a collection of open sets in T α for each α . Hence their union S U β is in T α for each α , and so S U β ∈ T . (3) Starting with a finite collection of open sets in T , the argument is as in (2) above....
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 Spring '05
 SMITH
 Topology, Sets, Empty set, Metric space, Topological space, U. By

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