122 5.2 Continuity and Compactness Theorem 5.1.12 Let D be a subset of R. The following hold: (i) The subsets 0/ and D are closed in D. (ii) The intersection of any collection of closed sets in D is closed in D. (iii) The union of a fnite number of closed sets in D is closed in D. 1 ■ Example 5.1.9 Consider the set D =[0,1) and the subset A =[2 ,1). Clearly, A is bounded. We showed in Example 5.1.8 that A is closed in D. However, A is not compact. We show this by fnding a sequence {an} in A for which no subsequence converges to a point in A. Indeed, consider the sequence an = 1 − 1 for n ∈ N. Then an ∈ A for all n. Moreover, {an} 2n converges to 1 and, hence, every subsequence also converges to 1. Since 1 ̸ ∈ A, it follows that A is not compact. Exercises 5.1.1 Prove that a subset A of R is open if and only if for any x ∈ A, there exists n ∈ N such that (x − 1/n,x + 1/n) ⊂ A. 5.1.2 Prove that the interval [0,1) is neither open nor closed. 5.1.3 ▶ Prove that if A and B are compact subsets of R, then A∪ B is a compact set. 5.1.4 Prove that any fnite set is compact. (Hint: frst prove that a set with a single element is compact and then use Exercise 5.1.3.) 5.1.5 Prove that the intersection of any collection of compact subsets of R is compact. 5.1.6 Find all limit points and all isolated points of each of the following sets: (a) A =(0,1). (b) B =[0,1). (c) C = Q. (d) D = {m + 1/n : m,n ∈ N}. 5.1.7 Let S =[0,∞). Classify each subset of S below as open in S, closed in S, neither or both. Justify your answers. (a) A =(0,1). (b) B = N. (c) C = Q ∩ A. (d) D =(−1,1]. (e) E =(−2,∞). 5.2 Continuity and Compactness Recall from Defnition 5.1.3 that a subset A of R is compact if and only if every sequence {an} in A has a subsequence {ank } that converges to a point a ∈ A.
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