16 1. THE CONSTRUCTION OF REAL AND COMPLEX NUMBERS Definition 1.6.22. Let S be a subset of R. We say that S is an open set in R if, for each point x ∈ S, there is an ε 0 (depending on x) such that (x − ε, x + ε) ⊆ S. Definition 1.6.23. Let S ⊆ R. We say S is a closed set in R if the complement of S is an open set in R. Note that the empty set and R are both open and closed subsets of R. Exercise 1.6.24. (i) Show that ∅ and R are the only subsets of R which are both open and closed in R. (ii) Show that every nonempty open set in R can be written as a countable union of pairwise disjoint open intervals. (iii) Show that an arbitrary union of open sets in R is open in R. (iv) Show that a finite intersection of open sets in R is open in R. (v) Show, by example, that an infinite intersection of open sets is not necessarily open. (vi) Show that an arbitrary intersection of closed sets in R is a closed set in R. (vii) Show that a finite union of closed sets in R is a closed set in R. (viii) Show, by example, that an infinite union of closed sets in R is not necessarily a closed set in R. Exercise 1.6.25. Show that a subset of R is closed iff it contains all its accumulation points. Exercise 1.6.26. In this exercise, we define the Cantor set. This is a subset of the closed interval [0, 1] constructed as follows. First, remove the open interval (1/3, 2/3) from [0, 1]. Next, remove the open intervals (1/9, 2/9) and (7/9, 8/9). At each step, remove the middle third of the remaining closed intervals. Repeating this process a countable number of times, we are left with a subset of the closed interval [0, 1] called the Cantor set. Show that: (i) the Cantor set is closed (ii) the Cantor set is uncountable (iii) the Cantor set consists of all numbers in the closed interval [0, 1] whose ternary expansion consists of only 0’s and 2’s and may end in infinitely many 2’s (iv) every point of the Cantor set is an accumulation point of the Cantor set (v) the set [0, 1] \ {Cantor set} is a dense subset of [0, 1]. The next theorem, the Heine-Borel theorem for R, is the second of the two basic topological theorems for the real numbers the other is the Bolzano- Weierstrass theorem. We shall see more details about these two theorems in Chapter 2.

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