Answer: For the following statement S and proposed proof, either (1) S is true and the

Chapter 16, Problem 16.27

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For the following statement S and proposed proof, either (1) S is true and the proof iscorrect, (2) S is true and the proof is incorrect, or (3) S is false and the proof is incorrect.Explain which of these occurs.S: Let X be an infinite set and let consists of and all infinite subsets of X. Then(X, ) is a topological space.Proof. Since X is an infinite subset of X, it follows that X . Since , property(1) of a topological space is satisfied. Let O1, O2,...,On be elements of for n N.We show that ni=1Oi . If Oi = for some i with 1 i n, then ni=1Oi = andni=1Oi . Otherwise, Oi is infinite for all i ( i n). Hence ni=1Oi is infinite and soni=1Oi . Thus property (2) is satisfied. Next, let {O}I be an indexed family ofopen sets. If O = for each I, then IO = and so IO . Otherwise, Ois infinite for some I and so IO is infinite. Hence IO . Therefore, is atopology on X.