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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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• Anuj s.
notation means index set
• Shashi ranjan sinha Best Answer
The encircled notion is of index set. The index set may be finite or infinite, it can be some symbols. Usually, for arbitrary family of something, we take natural numbers as our in...