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7. (a) Let A C X,B C X where X is a metric space. Show AnB CĀ OĒ. (b) Consider A = [3,7),B = (7,9] as subset of IR with the usual metric, dr (x, y) = | x – y| . Show Ā n B is not a subset of AnB

7. (a) Let A C X,B C X where X is a metric space. Show AnB CĀ OĒ. (b) Consider A = [3,7),B = (7,9] as subset of IR with the usual metric, dr (x, y) = | x – y| . Show Ā n B is not a subset of AnB

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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--- ## Metrics and Compactness in Metric Spaces ### 1. Determine if the following functions are metrics on \(\mathbb{R}\): Given \(x, y \in \mathbb{R}\): **(a)** \(d_1(x, y) = \sqrt{|x - y|}\) **(b)** \(d_2(x, y) = (x - y)^4\) ### 2. Interior Points and Openness Let \(E \subseteq X\) where \(X\) is a metric space. Let \(E^\circ\) denote all the interior points of \(E\). **Note:** \(E^\circ\) is an open set—you don’t have to prove this, but you can use this fact to prove the following. #### (a) Show that \(E\) is open if and only if \(E = E^\circ\). #### (b) Prove that 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 \(X\). Prove that \( K_1 \cup K_2 \) is compact. ### 4. Collection of Compact Subsets Let \( \{ K_\alpha \}_{\alpha \in A} \) be a collection of compact subsets of a metric space \(X\). Show that \( \cap_{\alpha \in A} K_\alpha \) is closed in \(X\). ### 5. Open Sets and Compactness in \(\mathbb{R}\) #### (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 \(\mathbb{R} = \cup_{n \in \mathbb{N}} (-n, n)\) as part of your work. #### (b) Give an example of a collection of compact sets in \(\mathbb{R}\), say \( \{K
Transcribed Image Text:--- ## Metrics and Compactness in Metric Spaces ### 1. Determine if the following functions are metrics on \(\mathbb{R}\): Given \(x, y \in \mathbb{R}\): **(a)** \(d_1(x, y) = \sqrt{|x - y|}\) **(b)** \(d_2(x, y) = (x - y)^4\) ### 2. Interior Points and Openness Let \(E \subseteq X\) where \(X\) is a metric space. Let \(E^\circ\) denote all the interior points of \(E\). **Note:** \(E^\circ\) is an open set—you don’t have to prove this, but you can use this fact to prove the following. #### (a) Show that \(E\) is open if and only if \(E = E^\circ\). #### (b) Prove that 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 \(X\). Prove that \( K_1 \cup K_2 \) is compact. ### 4. Collection of Compact Subsets Let \( \{ K_\alpha \}_{\alpha \in A} \) be a collection of compact subsets of a metric space \(X\). Show that \( \cap_{\alpha \in A} K_\alpha \) is closed in \(X\). ### 5. Open Sets and Compactness in \(\mathbb{R}\) #### (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 \(\mathbb{R} = \cup_{n \in \mathbb{N}} (-n, n)\) as part of your work. #### (b) Give an example of a collection of compact sets in \(\mathbb{R}\), say \( \{K
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