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1. Two metrics d₁ and d₂ on the same set X are said to be topologically equivalent if they generate the same topology on X. It can be proved that d₁ and d₂ are topologically equivalent if and only if for every point x € X and any radius ro > 0, we can find r₁ > 0 and r₂ > 0 such that Br₁(x, d₁) ≤ Bro (x, d₂) and Br₂(x, d₂) ≤ Bro (x, d₁). Let d₁((x₁, y₁), (x2, Y2)) = √√√(x1 − x2)² + (y₁ − y2)² be the Euclidean metric on X = R², and d₂((x₁, y₁), (x2, Y2)) = max{|x₁ — X2|, |Y1 — Y2|} be another metric on the same set R². Show that d₁ and d₂ are topologically equivalent.

1. Two metrics d₁ and d₂ on the same set X are said to be topologically equivalent if they generate the same topology on X. It can be proved that d₁ and d₂ are topologically equivalent if and only if for every point x € X and any radius ro > 0, we can find r₁ > 0 and r₂ > 0 such that Br₁(x, d₁) ≤ Bro (x, d₂) and Br₂(x, d₂) ≤ Bro (x, d₁). Let d₁((x₁, y₁), (x2, Y2)) = √√√(x1 − x2)² + (y₁ − y2)² be the Euclidean metric on X = R², and d₂((x₁, y₁), (x2, Y2)) = max{|x₁ — X2|, |Y1 — Y2|} be another metric on the same set R². Show that d₁ and d₂ are topologically equivalent.

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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1. Two metrics d₁ and d₂ on the same set X are said to be topologically equivalent if they generate the same topology on X. It can be proved that di and d₂ are topologically equivalent if and only if for every point x = X and any radius ro > 0, we can find r₁ > 0 and r₂ > 0 such that Br₁(x, d₁) Bro (x, d₂) and Br₂(x, d₂) ≤ Bro (x, d₁). Let d₁((x₁, y₁), (x2, Y2)) = √√√(x1 − x2)² + (y1 − y2)² be the Euclidean metric on X R2, - - and d₂((x₁, y₁), (x2, Y2)) = max{|x₁ - x2|, |y₁ - y2|} be another metric on the same set R². Show that di and d2 are topologically equivalent. =
Transcribed Image Text:1. Two metrics d₁ and d₂ on the same set X are said to be topologically equivalent if they generate the same topology on X. It can be proved that di and d₂ are topologically equivalent if and only if for every point x = X and any radius ro > 0, we can find r₁ > 0 and r₂ > 0 such that Br₁(x, d₁) Bro (x, d₂) and Br₂(x, d₂) ≤ Bro (x, d₁). Let d₁((x₁, y₁), (x2, Y2)) = √√√(x1 − x2)² + (y1 − y2)² be the Euclidean metric on X R2, - - and d₂((x₁, y₁), (x2, Y2)) = max{|x₁ - x2|, |y₁ - y2|} be another metric on the same set R². Show that di and d2 are topologically equivalent. =
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