Definition 4.3.9. Let (S, d) be a metric space. We say that U is an open subset of S provided Int(U) =U. That is, U is an open subset provided for any a E U , there exists an Ez > 0 such that B(x, Ez) C U.The notation E, reminds us that the radius of the ball centered at æ that is contained in U will dependon x.Which sets are open in the metric space (IR, d), where d(x, y) = |x – y|? Explain by appealing to denition 4.3.9.i). [1, 2)ii). (1, 2) U (2, 3)

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Which sets are open in the metric space (IR, d), v i). [1, 2) ii). (1, 2) U (2, 3)

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter8: Applications Of Trigonometry

Section8.3: Vectors

Problem 60E

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Question

Definition 4.3.9. Let (S, d) be a metric space. We say that U is an open subset of S provided Int(U) =
U. That is, U is an open subset provided for any a E U , there exists an Ez > 0 such that B(x, Ez) C U.
The notation E, reminds us that the radius of the ball centered at æ that is contained in U will depend
on x.
Which sets are open in the metric space (IR, d), where d(x, y) = |x – y|? Explain by appealing to denition 4.3.9.
i). [1, 2)
ii). (1, 2) U (2, 3)

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Answer

ii). (1, 2) U (2, 3)

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