120 5.1 Topology of the Real Line Defnition 5.1.5 Let D be a subset of R. We say that a subset V of D is open in D if for every a ∈ V, there exists δ > 0 such that B(a;δ ) ∩ D ⊂ V. Theorem 5.1.8 Let D be a subset of R. A subset V of D is open in D if and only if there exists an open subset G of R such that V = D ∩ G. Proof: Suppose V is open in D. By defnition, for every a ∈ V, there exists δa > 0 such that B(a;δa) ∩ D ⊂ V. Defne [ G = B(a;δa) a∈V Then G is a union of open subsets of R, so G is open. Moreover, [ V ⊂ G ∩ D = [B(a;δa) ∩ D] ⊂ V. a∈V Therefore, V = G ∩ D. Let us now prove the converse. Suppose V = G ∩ D, where G is an open set. For any a ∈ V, we have a ∈ G, so there exists δ > 0 such that B(a;δ ) ⊂ G. It follows that B(a;δ ) ∩ D ⊂ G ∩ D = V. The proof is now complete. □ 1 1 1 ■ Example 5.1.7 Let D =[0,1) and V =[0, 2 ). We can write V = D ∩ (−1, 2 ). Since (−1, 2 ) is open in R, we conclude from Theorem 5.1.8 that V is open in D. Notice that V itself is not an open subset of R. The following theorem is now a direct consequence of Theorems 5.1.8 and 5.1.1. Theorem 5.1.9 Let D be a subset of R. The following hold: (i) The subsets 0/ and D are open in D. (ii) The union of any collection of open sets in D is open in D. (iii) The intersection of a fnite number of open sets in D is open in D. Defnition 5.1.6 Let D be a subset of R. We say that a subset A of D is closed in D if D \ A is open in D.
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