Let K and L be nonempty compact sets, and define d = inf{|r - y|:1€ K and y E L} This turns out to be a reasonable definition for the distance between K and L. (a) If K and L are disjoint, show d > 0 and that d = |r0 – y0| for some ro € K and yo E L. (b) Show that it's possible to have d = 0 if we assume only that the disjoint sets K and L are closed.

Let K and L be nonempty compact sets, and define d = inf{|r - y|:1€ K and y E L} This turns out to be a reasonable definition for the distance between K and L. (a) If K and L are disjoint, show d > 0 and that d = |r0 – y0| for some ro € K and yo E L. (b) Show that it's possible to have d = 0 if we assume only that the disjoint sets K and L are closed.

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter5: Rings, Integral Domains, And Fields

Section5.4: Ordered Integral Domains

Problem 1E: Complete the proof of Theorem 5.30 by providing the following statements, where and are arbitrary...

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