Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN: 9780470458365
Author: Erwin Kreyszig
Publisher: Wiley, John & Sons, Incorporated
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Here is the transcription of the image for use on an educational website: --- ### Real Analysis and Metric Spaces Problem Set #### 1. Metric Determination Let \( x, y \in \mathbb{R} \). Determine if the following are metrics on \(\mathbb{R}\) or not: - (a) \(d_1(x, y) = \sqrt{|x - y|}\) - (b) \(d_2(x, y) = (x - y)^4\) #### 2. Interior Points and Open Sets Let \( E \subseteq X \) where \( X \) is a metric space. Let \( E^\circ \) denote all the interior points of \( E \). \( E^\circ \) is an open set—you don’t have to prove this, but you can use this fact to prove the following: - (a) Show \( E \) is open if and only if \( E = E^\circ \). - (b) Show: If \( G \subseteq E \) and \( G \) is open, then \( G \subseteq E^\circ \). #### 3. Compact Subsets Let \( K_1 \) and \( K_2 \) be compact subsets of a metric space \( X \). Show \( K_1 \cup K_2 \) is compact. #### 4. Closedness of Intersection of Compact Sets Let \(\{K_\alpha\}_{\alpha \in A}\) be a collection of compact subsets of a metric space \( X \). Show \(\bigcap_{\alpha \in A} K_\alpha\) is closed in \( X \). #### 5. Non-compactness of Open and Compact Sets - (a) Consider the collection of open sets, \(\{(-n, n)\}_{n=1}^\infty\) in \(\mathbb{R}\) with the usual metric \( d(x, y) = |x - y| \). Use this collection of open sets to show \(\mathbb{R}\) is not compact (make sure to prove \(\mathbb{R} = \bigcup_{n \in \mathbb{N}} (-n, n)\) as part of your work). - (b) Give an example of a collection of compact sets in
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Transcribed Image Text:Here is the transcription of the image for use on an educational website: --- ### Real Analysis and Metric Spaces Problem Set #### 1. Metric Determination Let \( x, y \in \mathbb{R} \). Determine if the following are metrics on \(\mathbb{R}\) or not: - (a) \(d_1(x, y) = \sqrt{|x - y|}\) - (b) \(d_2(x, y) = (x - y)^4\) #### 2. Interior Points and Open Sets Let \( E \subseteq X \) where \( X \) is a metric space. Let \( E^\circ \) denote all the interior points of \( E \). \( E^\circ \) is an open set—you don’t have to prove this, but you can use this fact to prove the following: - (a) Show \( E \) is open if and only if \( E = E^\circ \). - (b) Show: If \( G \subseteq E \) and \( G \) is open, then \( G \subseteq E^\circ \). #### 3. Compact Subsets Let \( K_1 \) and \( K_2 \) be compact subsets of a metric space \( X \). Show \( K_1 \cup K_2 \) is compact. #### 4. Closedness of Intersection of Compact Sets Let \(\{K_\alpha\}_{\alpha \in A}\) be a collection of compact subsets of a metric space \( X \). Show \(\bigcap_{\alpha \in A} K_\alpha\) is closed in \( X \). #### 5. Non-compactness of Open and Compact Sets - (a) Consider the collection of open sets, \(\{(-n, n)\}_{n=1}^\infty\) in \(\mathbb{R}\) with the usual metric \( d(x, y) = |x - y| \). Use this collection of open sets to show \(\mathbb{R}\) is not compact (make sure to prove \(\mathbb{R} = \bigcup_{n \in \mathbb{N}} (-n, n)\) as part of your work). - (b) Give an example of a collection of compact sets in
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