MATH_3333_HW3_solns - Math 3333 Homework 3 NAME(Print 1 TRUEFALSE If your answer is True justify by quoting a definition or theorem or by giving a proof

# MATH_3333_HW3_solns - Math 3333 Homework 3 NAME(Print 1...

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Math 3333 Homework 3 NAME (Print): 1. TRUE–FALSE. If your answer is “True,” justify by quoting a definition ortheorem, or by giving a proof. If your answer is “False,” give a counter-example.(a) LetGbe a collection of closed sets. Then\Ais closed. A ∈G TRUE: \ A ∈G A ! c = [ A ∈G A c . Since A is closed for all A ∈ G , A c is open for all A . Since the union of any collection of open sets is open, [ A ∈G A c = \ A ∈G A ! c is open. Therefore, \ A ∈G A is closed. (b) Let F be a collection of closed sets. Then [ B ∈G B is closed. FALSE: Let B 1 = 1 3 , 2 3 , B 2 = 1 4 , 3 4 , B 3 = 1 5 , 4 5 , . . . , B n = 1 n , n - 1 n , . . . . [ n =1 B n = (0 , 1) which is open. 1 (c) Ifpis an isolated point ofS,thenpis a boundary point ofS.  • • • 