Exercise 4. Suppose K is a compact subset of the metric space (X, d). a) Show that for each x X, there is a point k such that d(x, kx) = dk(x) = d(x, K) = info d(k, x). b) Endow R² with the metric dmax. If K = [−1, 1] × {0} C R², find a point (x, y) = R² such that dmax ((x, y), K) = dmax ((x, y), (t,0)), for all t € [−1,1]. c) Endow R² with the Euclidean metric metric d₂. If K = [−1, 1] × {0} ℃ R², show that for every (x, y) = R² there is a unique t € [-1,1] with d₂((x, y), K) = d₂((x, y), (t,0)).

Exercise 4. Suppose K is a compact subset of the metric space (X, d). a) Show that for each x X, there is a point k such that d(x, kx) = dk(x) = d(x, K) = info d(k, x). b) Endow R² with the metric dmax. If K = [−1, 1] × {0} C R², find a point (x, y) = R² such that dmax ((x, y), K) = dmax ((x, y), (t,0)), for all t € [−1,1]. c) Endow R² with the Euclidean metric metric d₂. If K = [−1, 1] × {0} ℃ R², show that for every (x, y) = R² there is a unique t € [-1,1] with d₂((x, y), K) = d₂((x, y), (t,0)).

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter3: Functions And Graphs

Section3.2: Graphs Of Equations

Problem 67E

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