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The collection {a} × (b, c) C R² | a, b, c E R} of vertical intervals in the plane is a basis for a topology on IR². We call this topology the vertical interval topology.

The collection {a} × (b, c) C R² | a, b, c E R} of vertical intervals in the plane is a basis for a topology on IR². We call this topology the vertical interval topology.

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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The collection \(\{a\} \times (b, c) \subseteq \mathbb{R}^2 \mid a, b, c \in \mathbb{R}\) of vertical intervals in the plane is a basis for a topology on \(\mathbb{R}^2\). We call this topology the **vertical interval topology**.
Transcribed Image Text:The collection \(\{a\} \times (b, c) \subseteq \mathbb{R}^2 \mid a, b, c \in \mathbb{R}\) of vertical intervals in the plane is a basis for a topology on \(\mathbb{R}^2\). We call this topology the **vertical interval topology**.
**Problem Statement:** Show that the collection \(\{a\} \times (b, c) \subseteq \mathbb{R}^2 \mid a, b, c \in \mathbb{R}\) of vertical intervals in the plane is a basis for a topology on \(\mathbb{R}^2\). **Explanation:** This problem asks us to prove that a specific collection of vertical line segments (of the form \(\{a\} \times (b, c)\)) can generate a topology on the Euclidean plane \(\mathbb{R}^2\). Each element in the collection is defined by fixing a real number \(a\) and considering intervals \((b, c)\) along the vertical axis. These intervals are subsets of \(\mathbb{R}^2\). A topology on \(\mathbb{R}^2\) involves defining a set of open sets that satisfy certain criteria, and showing that this collection can serve as a basis for such a topology involves verifying that it meets the requirements of a topology basis: 1. **Covering Requirement:** For each point in \(\mathbb{R}^2\), there exists an element of the basis that contains it. 2. **Intersection Requirement:** If a point belongs to the intersection of two basis elements, there exists another basis element that contains that point and is contained within the intersection of the two basis elements.
Transcribed Image Text:**Problem Statement:** Show that the collection \(\{a\} \times (b, c) \subseteq \mathbb{R}^2 \mid a, b, c \in \mathbb{R}\) of vertical intervals in the plane is a basis for a topology on \(\mathbb{R}^2\). **Explanation:** This problem asks us to prove that a specific collection of vertical line segments (of the form \(\{a\} \times (b, c)\)) can generate a topology on the Euclidean plane \(\mathbb{R}^2\). Each element in the collection is defined by fixing a real number \(a\) and considering intervals \((b, c)\) along the vertical axis. These intervals are subsets of \(\mathbb{R}^2\). A topology on \(\mathbb{R}^2\) involves defining a set of open sets that satisfy certain criteria, and showing that this collection can serve as a basis for such a topology involves verifying that it meets the requirements of a topology basis: 1. **Covering Requirement:** For each point in \(\mathbb{R}^2\), there exists an element of the basis that contains it. 2. **Intersection Requirement:** If a point belongs to the intersection of two basis elements, there exists another basis element that contains that point and is contained within the intersection of the two basis elements.
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