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Theorem 9.13. Let (X, d) and (Y,e) be metric spaces. Then X × Y is a metric space.

Theorem 9.13. Let (X, d) and (Y,e) be metric spaces. Then X × Y is a metric space.

Advanced Engineering Mathematics
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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Could you explain how to show 9.13 in detail?

**Theorem 9.13.** Let \((X, d)\) and \((Y, e)\) be metric spaces. Then \(X \times Y\) is a metric space. *Explanation:* This theorem describes a property of metric spaces in topology and geometry. A metric space is a set equipped with a function (called a metric) that defines a distance between any two elements in the set. - \(X\) and \(Y\) are sets, and \(d\) and \(e\) are metrics on these sets, respectively. - \(X \times Y\) represents the Cartesian product of the sets \(X\) and \(Y\), which is the set of all ordered pairs \((x, y)\) where \(x \in X\) and \(y \in Y\). The theorem states that when you take two metric spaces and form the Cartesian product of their sets, the resulting set can also be considered a metric space.
Transcribed Image Text:**Theorem 9.13.** Let \((X, d)\) and \((Y, e)\) be metric spaces. Then \(X \times Y\) is a metric space. *Explanation:* This theorem describes a property of metric spaces in topology and geometry. A metric space is a set equipped with a function (called a metric) that defines a distance between any two elements in the set. - \(X\) and \(Y\) are sets, and \(d\) and \(e\) are metrics on these sets, respectively. - \(X \times Y\) represents the Cartesian product of the sets \(X\) and \(Y\), which is the set of all ordered pairs \((x, y)\) where \(x \in X\) and \(y \in Y\). The theorem states that when you take two metric spaces and form the Cartesian product of their sets, the resulting set can also be considered a metric space.
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