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Math 3333Homework 3NAME (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∈GTRUE:\A∈GA!c=[A∈GAc. SinceAis closed for allA∈ G, Acisopen for allA.Since the union of any collection of open sets is open,[A∈GAc=\A∈GA!cis open. Therefore,\A∈GAis closed.(b) LetFbe a collection of closed sets. Then[B∈GBis closed.FALSE:LetB1=13,23, B2=14,34, B3=15,45, . . . , Bn=1n,n-1n, . . ..∞[n=1Bn= (0,1)which is open.1
(c) Ifpis an isolated point ofS,thenpis a boundary point ofS.