Introduction to Mathematical Analysis I - Second Edition

55 2.6 OPEN SETS, CLOSED SETS, COMPACT SETS, AND LIMIT POINTS The open ball in R with center a ∈ R and radius δ > 0 is the set B ( a ; δ ) = ( a − δ , a + δ ) . Definition 2.6.1 A subset A of R is said to be open if for each a ∈ A , there exists δ > 0 such that B ( a ; δ ) ⊂ A . Example 2.6.1 (1) Any open interval A = ( c , d ) is open. Indeed, for each a ∈ A , one has c < a < d . Let δ = min { a − c , d − a } . Then B ( a ; δ ) = ( a − δ , a + δ ) ⊂ A . Therefore, A is open. (2) The sets A = ( − ∞ , c ) and B = ( c , ∞ ) are open, but the set C = [ c , ∞ ) is not open. The reader can easily verify that A and B are open. Let us show that C is not open. Assume by contradiction that C is open. Then, for the element c ∈ C , there exists δ > 0 such that B ( c ; δ ) = ( c − δ , c + δ ) ⊂ C . However, this is a contradiction because c − δ / 2 ∈ B ( c ; δ ) , but c − δ / 2 / ∈ C . Theorem 2.6.1 The following hold: (a) The subsets /0 and R are open. (b) The union of any collection of open subsets of R is open. (c) The intersection of a finite number of open subsets of R is open. Proof: The proof of (a) is straightforward. (b) Suppose { G α : α ∈ I } is an arbitrary collection of open subsets of R . That means G α is open for every α ∈ I . Let us show that the set G = [ α ∈ I G α is open. Take any a ∈ G . Then there exists α 0 ∈ I such that a ∈ G α 0 . Since G α 0 is open, there exists δ > 0 such that B ( a ; δ ) ⊂ G α 0 . This implies B ( a ; δ ) ⊂ G because G α 0 ⊂ G . Thus, G is open.