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(a) Suppose M is a set and d, d' are two different metrics on M. Prove that d and d' generate the same topology on M if and only if the following condition is satisfied: for every x E M and every r > 0, there exist positive numbers ₁ and r2 such that Bd') (x) ≤ Bd) (x) and Bd) (x) Bd') (x).

(a) Suppose M is a set and d, d' are two different metrics on M. Prove that d and d' generate the same topology on M if and only if the following condition is satisfied: for every x E M and every r > 0, there exist positive numbers ₁ and r2 such that Bd') (x) ≤ Bd) (x) and Bd) (x) Bd') (x).

Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter3: Matrices
Section3.7: Applications
Problem 74EQ
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Exercise 2.4. (a) Suppose M is a set and d, d' are two different metrics on M. Prove that d and d' generate the same topology on M if and only if the following condition is satisfied: for every x E M and every r > 0, there exist positive numbers ₁ and r2 such that Bd (x) ≤ Bd) (x) and B)(x) ≤ Bd') (x). (b) Let (M, d) be a metric space, let c be a positive real number, and define a new metric
Transcribed Image Text:Exercise 2.4. (a) Suppose M is a set and d, d' are two different metrics on M. Prove that d and d' generate the same topology on M if and only if the following condition is satisfied: for every x E M and every r > 0, there exist positive numbers ₁ and r2 such that Bd (x) ≤ Bd) (x) and B)(x) ≤ Bd') (x). (b) Let (M, d) be a metric space, let c be a positive real number, and define a new metric
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