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Dhivya Dharshini Asked a Question October 11, 2020 6:05 pm 30 pts (7-8 Theorem: The union ot an aroltrary ramily of open sets is an onen cot Proof. Let (Gà e A} be an arbitrary family of open set. Let G = uG. We have to show that G is an open set. A Let x be any element of G = u G Then x e G, for some à e A. Since G. is given to be an open set and x e G, so G, is a nbd. of x. Thus there exist some a >0 such that XE (XE, X + E) C G xeX- , X + E) C G, CU G = G EA XE (x - s, X + E) CG V xe G Gis a nbd. of x, V x e G. t tollows that G is a nbd. of each of its points and consequently G = G, is an open set Theorem: The intersection of a finite number of open sets is an open set. 135 IColl Toll Free: 1800-120-1021

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