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

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

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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**Definition 4.3.9.** Let \((S, d)\) be a metric space. We say that \(U\) is an *open* subset of \(S\) provided \(\text{Int}(U) = U\). That is, \(U\) is an open subset provided for any \(x \in U\), there exists an \(\epsilon_x > 0\) such that \(B(x, \epsilon_x) \subseteq U\). The notation \(\epsilon_x\) reminds us that the radius of the ball centered at \(x\) that is contained in \(U\) will depend on \(x\). *Which sets are open in the metric space \((\mathbb{R}, d)\), where \(d(x, y) = |x - y|\)? Explain by appealing to Definition 4.3.9.* i) \([1, 2)\) ii) \((1, 2) \cup (2, 3)\)
Transcribed Image Text:**Definition 4.3.9.** Let \((S, d)\) be a metric space. We say that \(U\) is an *open* subset of \(S\) provided \(\text{Int}(U) = U\). That is, \(U\) is an open subset provided for any \(x \in U\), there exists an \(\epsilon_x > 0\) such that \(B(x, \epsilon_x) \subseteq U\). The notation \(\epsilon_x\) reminds us that the radius of the ball centered at \(x\) that is contained in \(U\) will depend on \(x\). *Which sets are open in the metric space \((\mathbb{R}, d)\), where \(d(x, y) = |x - y|\)? Explain by appealing to Definition 4.3.9.* i) \([1, 2)\) ii) \((1, 2) \cup (2, 3)\)
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ii). (1, 2) U (2, 3)

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