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3. Find an open cover of (x:x>0} with no finite subcover.

3. Find an open cover of (x:x>0} with no finite subcover.

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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**Problem 33**: Find an open cover of \(\{x : x > 0\}\) with no finite subcover. *Explanation*: The task is to construct an open cover for the set of positive real numbers, \(\{x : x > 0\}\), such that no finite subset of this cover can also serve as a cover for the same set. To achieve this, consider the set of open intervals \( \left(\frac{1}{n}, n\right) \) for all natural numbers \( n \). Each interval \( \left(\frac{1}{n}, n\right) \) is an open set that contributes to the cover of the set \(\{x : x > 0\}\). - The union of all such intervals as \( n \) ranges over the natural numbers covers all positive real numbers because for any positive \( x \), you can find a natural number \( n \) such that \( x \) falls within the interval \( \left(\frac{1}{n}, n\right) \). - If we take only a finite subset of this cover, say \( \left(\frac{1}{n_1}, n_1\right), \left(\frac{1}{n_2}, n_2\right), \ldots, \left(\frac{1}{n_k}, n_k\right) \), then there will be positive real numbers less than \( \frac{1}{n_m} \) for some maximum \( n_m \) that are not covered. Therefore, this provides an illustration of an open cover with no finite subcover, demonstrating the nature of the set of positive real numbers.
Transcribed Image Text:**Problem 33**: Find an open cover of \(\{x : x > 0\}\) with no finite subcover. *Explanation*: The task is to construct an open cover for the set of positive real numbers, \(\{x : x > 0\}\), such that no finite subset of this cover can also serve as a cover for the same set. To achieve this, consider the set of open intervals \( \left(\frac{1}{n}, n\right) \) for all natural numbers \( n \). Each interval \( \left(\frac{1}{n}, n\right) \) is an open set that contributes to the cover of the set \(\{x : x > 0\}\). - The union of all such intervals as \( n \) ranges over the natural numbers covers all positive real numbers because for any positive \( x \), you can find a natural number \( n \) such that \( x \) falls within the interval \( \left(\frac{1}{n}, n\right) \). - If we take only a finite subset of this cover, say \( \left(\frac{1}{n_1}, n_1\right), \left(\frac{1}{n_2}, n_2\right), \ldots, \left(\frac{1}{n_k}, n_k\right) \), then there will be positive real numbers less than \( \frac{1}{n_m} \) for some maximum \( n_m \) that are not covered. Therefore, this provides an illustration of an open cover with no finite subcover, demonstrating the nature of the set of positive real numbers.
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