Exercise 9. Closed product topology] Let X and Y be topologic al space, A. U be subsets of X, and B be a subset of Y. 1. Show that if A is closed in X and B is closed in Y then A x B is closed in X x Y. 2. Show that if U is open in X and A is closed in X, then U\A is open in X, and A\U is closed in X. 3. Prove that Ax B=Ax B.
Exercise 9. Closed product topology] Let X and Y be topologic al space, A. U be subsets of X, and B be a subset of Y. 1. Show that if A is closed in X and B is closed in Y then A x B is closed in X x Y. 2. Show that if U is open in X and A is closed in X, then U\A is open in X, and A\U is closed in X. 3. Prove that Ax B=Ax B.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
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exercice 9 topology
![**Exercise 7: Basis and Comparable Topology**
Let \( X = \mathbb{R} \) and \( K = \{\frac{1}{n} ; n \in \mathbb{N}\} \). Consider the following collections on \( X \):
- \( \mathcal{B} = \{[a, b]; a, b \in \mathbb{R}, a < b\} \)
- \( \mathcal{B}' = \{[a, b); a, b \in \mathbb{R}, a < b\} \)
- \( \mathcal{B}'' = \{[a, b); a, b \in \mathbb{R}, a < b\} \cup \{[a, b)(K; a, b \in \mathbb{R}, a < b)\} \)
Knowing that \( \mathcal{B} \) and \( \mathcal{B}' \) are bases for some topology on \( X \), prove that \( \mathcal{B}'' \) is a basis for a topology on \( X \). Furthermore, let \( \tau \), \( \tau' \), and \( \tau'' \) denote the topologies on \( X \) generated by \( \mathcal{B} \), \( \mathcal{B}' \), and \( \mathcal{B}'' \), respectively. Prove that \( \tau' \) and \( \tau'' \) are finer than \( \tau \), and that \( \tau' \) and \( \tau'' \) are not comparable.
**Exercise 8: Subspaces Product Topology**
Let \( A \) be a subspace of \( X \) and let \( B \) be a subspace of \( Y \). We equip \( A \) and \( B \) with the subspace topologies. Prove that the product topology on \( A \times B \) is the same as the topology \( A \times B \) inherits as a subspace of \( X \times Y \).
**Exercise 9: Closed Product Topology**
Let \( X \) and \( Y \) be topological spaces, \( A, U \) be subsets of \( X \), and \( B \) be a subset of \( Y \).
1. Show that](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3b5f7db1-a103-4627-9a7f-8775edb35b3d%2F6da5205f-4d61-41d2-9891-e0501fa9ad63%2Fbvk02nw_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Exercise 7: Basis and Comparable Topology**
Let \( X = \mathbb{R} \) and \( K = \{\frac{1}{n} ; n \in \mathbb{N}\} \). Consider the following collections on \( X \):
- \( \mathcal{B} = \{[a, b]; a, b \in \mathbb{R}, a < b\} \)
- \( \mathcal{B}' = \{[a, b); a, b \in \mathbb{R}, a < b\} \)
- \( \mathcal{B}'' = \{[a, b); a, b \in \mathbb{R}, a < b\} \cup \{[a, b)(K; a, b \in \mathbb{R}, a < b)\} \)
Knowing that \( \mathcal{B} \) and \( \mathcal{B}' \) are bases for some topology on \( X \), prove that \( \mathcal{B}'' \) is a basis for a topology on \( X \). Furthermore, let \( \tau \), \( \tau' \), and \( \tau'' \) denote the topologies on \( X \) generated by \( \mathcal{B} \), \( \mathcal{B}' \), and \( \mathcal{B}'' \), respectively. Prove that \( \tau' \) and \( \tau'' \) are finer than \( \tau \), and that \( \tau' \) and \( \tau'' \) are not comparable.
**Exercise 8: Subspaces Product Topology**
Let \( A \) be a subspace of \( X \) and let \( B \) be a subspace of \( Y \). We equip \( A \) and \( B \) with the subspace topologies. Prove that the product topology on \( A \times B \) is the same as the topology \( A \times B \) inherits as a subspace of \( X \times Y \).
**Exercise 9: Closed Product Topology**
Let \( X \) and \( Y \) be topological spaces, \( A, U \) be subsets of \( X \), and \( B \) be a subset of \( Y \).
1. Show that
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