Prove the following: Let X denote the set {0,1} with the discrete topology. Let Y = X" = || X; ieZ+ where X; = X for each i. Prove that the product topology on Y is not the discrete topology.
Prove the following: Let X denote the set {0,1} with the discrete topology. Let Y = X" = || X; ieZ+ where X; = X for each i. Prove that the product topology on Y is not the discrete topology.
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 Statement:**
Prove the following: Let \( X \) denote the set \(\{0,1\}\) with the discrete topology. Let
\[ Y = X^\omega = \prod_{i \in \mathbb{Z}_+} X_i \]
where \( X_i = X \) for each \( i \). Prove that the product topology on \( Y \) is not the discrete topology.
**Explanation:**
This mathematical problem involves proving that the product topology on an infinite product of discrete spaces is not discrete. Here, \( X \) is a discrete space with two elements, 0 and 1. The set \( Y \) is an infinite product of copies of \( X \), indexed by the positive integers \( \mathbb{Z}_+ \).
When dealing with topologies, the discrete topology means that every subset is open. In contrast, the product topology on an infinite product is usually coarser (i.e., has fewer open sets) than the discrete topology, unless the index set is finite. This is because only finite intersections of basic open sets (products where all but finitely many factors are the whole space) need to be in the topology.
The problem requires understanding the distinction between these topologies and exploiting properties of infinite products to demonstrate that the set cannot have the discrete topology.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F45acd104-f51d-4f09-aed7-d7503d466014%2Fbe62586b-d577-4c3f-a4fe-bb21e358be7b%2Fbotmrr8_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Prove the following: Let \( X \) denote the set \(\{0,1\}\) with the discrete topology. Let
\[ Y = X^\omega = \prod_{i \in \mathbb{Z}_+} X_i \]
where \( X_i = X \) for each \( i \). Prove that the product topology on \( Y \) is not the discrete topology.
**Explanation:**
This mathematical problem involves proving that the product topology on an infinite product of discrete spaces is not discrete. Here, \( X \) is a discrete space with two elements, 0 and 1. The set \( Y \) is an infinite product of copies of \( X \), indexed by the positive integers \( \mathbb{Z}_+ \).
When dealing with topologies, the discrete topology means that every subset is open. In contrast, the product topology on an infinite product is usually coarser (i.e., has fewer open sets) than the discrete topology, unless the index set is finite. This is because only finite intersections of basic open sets (products where all but finitely many factors are the whole space) need to be in the topology.
The problem requires understanding the distinction between these topologies and exploiting properties of infinite products to demonstrate that the set cannot have the discrete topology.
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