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Theorem 9.6. For any metric space (X, d), there exists a metric d such that d and d generate the same topology, yet for each x, y E X, d(x,y) < 1.

Theorem 9.6. For any metric space (X, d), there exists a metric d such that d and d generate the same topology, yet for each x, y E X, d(x,y) < 1.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
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Could you explain how to show 9.6 in detail?

Theorem 9.6. For any metric space (X, d), there exists a metric d such that d and d generate the same topology, yet for each x, y E X, d(x,y) < 1. Definition. A metric on a set M is a function d : M x M → non-negative real numbers) such that for all a, b, c E M, these properties hold: R+ (where R, is the (1) d(a, b) > 0, with d(a, b) = 0 if and only if a = b; %3D (2) d(a, b) = d(b, a); (3) d(a, c) < d(a, b) + d(b, c). These three properties are often summarized by saying that a metric is positive defi- nite, symmetric, and satisfies the triangle inequality. A metric space (M, d) is a set M with a metric d. Example. The function d(x, y) = |x – y| is a metric on R. This measure of distance is the standard metric on R. Example. On any set M, we can define the discrete metric as follows: for any a, b e M, d(a, b) 1 if a + b and d(a, a) = 0. This metric basically tells us whether two points are the same or different. m Example. Here's a strange metric on Q: for reduced fractions, let d(, ") = max(|a – m|, |b – nl). Which rationals are "close" to one another under this metric? %3D n Theorem 9.3. Let d be a metric on the set X. Then the collection of all open balls В %3 В(р, е) 3 {у € X\d(p, у) < е} for every p € X and every e > 0} forms a basis for a topology on X. The topology generated by a metric d on X is called the d-metric topology for X. Definition. A topological space (X,T) is a metric space or is metrizable if and only if there is a metric d on X such that J is the d-metric topology. We sometimes write a metric space as (X, d) to denote X with the d-metric topology.
Transcribed Image Text:Theorem 9.6. For any metric space (X, d), there exists a metric d such that d and d generate the same topology, yet for each x, y E X, d(x,y) < 1. Definition. A metric on a set M is a function d : M x M → non-negative real numbers) such that for all a, b, c E M, these properties hold: R+ (where R, is the (1) d(a, b) > 0, with d(a, b) = 0 if and only if a = b; %3D (2) d(a, b) = d(b, a); (3) d(a, c) < d(a, b) + d(b, c). These three properties are often summarized by saying that a metric is positive defi- nite, symmetric, and satisfies the triangle inequality. A metric space (M, d) is a set M with a metric d. Example. The function d(x, y) = |x – y| is a metric on R. This measure of distance is the standard metric on R. Example. On any set M, we can define the discrete metric as follows: for any a, b e M, d(a, b) 1 if a + b and d(a, a) = 0. This metric basically tells us whether two points are the same or different. m Example. Here's a strange metric on Q: for reduced fractions, let d(, ") = max(|a – m|, |b – nl). Which rationals are "close" to one another under this metric? %3D n Theorem 9.3. Let d be a metric on the set X. Then the collection of all open balls В %3 В(р, е) 3 {у € X\d(p, у) < е} for every p € X and every e > 0} forms a basis for a topology on X. The topology generated by a metric d on X is called the d-metric topology for X. Definition. A topological space (X,T) is a metric space or is metrizable if and only if there is a metric d on X such that J is the d-metric topology. We sometimes write a metric space as (X, d) to denote X with the d-metric topology.
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