**Theorem 4.3.11.** Suppose \((S, d)\) is a metric space. Then the following sets are open:i). the set \(S\) itselfii). 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 \(\epsilon\).*

Consider the metric space S,d where S is the set of all real numbers and the metric d on S be…

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

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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