120 5.1 Topology of the Real Line Definition 5.1.5 Let Dbe a subset of R. We say that a subset V of Dis open in Dif for every a∈V, there exists δ >0 such that B(a;δ)∩D⊂V. Theorem 5.1.8 Let Dbe a subset of R. A subset V of Dis open inDif and only if there exists an open subset Gof Rsuch that V =D∩G. Proof: SupposeV is open in D. By definition, for everya∈V, there exists δa >0 such that B(a;δa)∩D⊂V. Define G= [ a∈V B(a;δa) Then Gis a union of open subsets of R, soGis open. Moreover, V ⊂G∩D= [ a∈V [B(a;δa)∩D] ⊂V. Therefore, V =G∩D. Let us now prove the converse. SupposeV =G∩D, where Gis 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. □ ■ Example 5.1.7 Let D= [0,1) andV = [0, 1 2). We can write V =D∩(−1, 1 2). Since (−1, 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 Dbe a subset of R. The following hold: (i) The subsets 0/ and Dare open inD. (ii) The union of any collection of open sets inDis open in D. (iii) The intersection of a finite number of open sets inDis open in D. Definition 5.1.6 Let Dbe a subset of R. We say that a subset Aof Dis closed in Dif D\Ais open inD.
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