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Theorem 4.3.11. Suppose (S, d) is a metric space. Then the following sets are open: i). the set S itself i). the empty set Ø ii), the open ball B(x, E) for any x and any E> 0 iv). UnV for any open sets U C S and VCS V). any finite or infinite union of open sets. That is U {U : U is open in S}is open Use part ii) of Theorem 4.3.11 to show the interval (0, 1) is open. That is, show (0, 1) can be written as B(x, e) for some x and e.

Theorem 4.3.11. Suppose (S, d) is a metric space. Then the following sets are open: i). the set S itself i). the empty set Ø ii), the open ball B(x, E) for any x and any E> 0 iv). UnV for any open sets U C S and VCS V). any finite or infinite union of open sets. That is U {U : U is open in S}is open Use part ii) of Theorem 4.3.11 to show the interval (0, 1) is open. That is, show (0, 1) can be written as B(x, e) for some x and e.

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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**Theorem 4.3.11.** Suppose \((S, d)\) is a metric space. Then the following sets are open: i). the set \(S\) itself ii). the empty set \(\emptyset\) iii). the open ball \(B(x, \epsilon)\) for any \(x\) and any \(\epsilon > 0\) iv). \(U \cap V\) for any open sets \(U \subseteq S\) and \(V \subseteq S\) v). any finite or infinite union of open sets. That is \( \bigcup \{ U : U \text{ is open in } S \} \) is open *Use part iii) of Theorem 4.3.11 to show the interval \((0, 1)\) is open. That is, show \((0, 1)\) can be written as \(B(x, \epsilon)\) for some \(x\) and \(\epsilon\).*
Transcribed Image Text:**Theorem 4.3.11.** Suppose \((S, d)\) is a metric space. Then the following sets are open: i). the set \(S\) itself ii). the empty set \(\emptyset\) iii). the open ball \(B(x, \epsilon)\) for any \(x\) and any \(\epsilon > 0\) iv). \(U \cap V\) for any open sets \(U \subseteq S\) and \(V \subseteq S\) v). any finite or infinite union of open sets. That is \( \bigcup \{ U : U \text{ is open in } S \} \) is open *Use part iii) of Theorem 4.3.11 to show the interval \((0, 1)\) is open. That is, show \((0, 1)\) can be written as \(B(x, \epsilon)\) for some \(x\) and \(\epsilon\).*
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