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 dgenerate 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 isthe standard metric on R.Example. On any set M, we can define the discrete metric as follows: for any a, b eM, d(a, b)1 if a + b and d(a, a)= 0. This metric basically tells us whether twopoints are the same or different.mExample. 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?%3DnTheorem 9.3. Let d be a metric on the set X. Then the collection of all open ballsВ %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 onlyif there is a metric d on X such that J is the d-metric topology. We sometimes write ametric space as (X, d) to denote X with the d-metric topology.

Let X, d be any metric space. It is required to prove that there exists a metric d such that d and…

Answered: Theorem 9.6. For any metric space (X,…

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.

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Find the average of 1/2, 1/3 , and1/4
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