122 5.2 Continuity and Compactness Theorem 5.1.12 Let Dbe a subset of R. The following hold: (i) The subsets 0/ and Dare closed in D. (ii) The intersection of any collection of closed sets inDis closed inD. (iii) The union of a finite number of closed sets inDis closed inD. ■ Example 5.1.9 Consider the set D= [0,1) and the subset A= [ 1 2,1). Clearly, Ais bounded. We showed in Example 5.1.8 that Ais closed in D. However, Ais not compact. We show this by finding a sequence {an}inAfor which no subsequence converges to a point in A. Indeed, consider the sequence an =1− 1 2n for n∈N. Then an ∈Afor all n. Moreover, {an} converges to 1 and, hence, every subsequence also converges to 1. Since 1̸∈A, it follows that Ais not compact. Exercises 5.1.1 Prove that a subset Aof Ris open if and only if for any x ∈A, there exists n∈Nsuch 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 AandBare compact subsets of R, then A∪Bis a compact set. 5.1.4 Prove that any finite set is compact. (Hint: first 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 Ris 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 Definition 5.1.3 that a subset Aof Ris compact if and only if every sequence {an} inAhas a subsequence {ank}that converges to a point a∈A.
RkJQdWJsaXNoZXIy NTc4NTAz