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Let A, B C R. We define the distance between A and B by d(A, B) = inf{|x - y| : x E A, y E B}. Show that if A is compact and B is closed, and d(A, B) = 0, then ANB#Ø. • Give examples of d(A, B) = 0, ANB=Ø when o A and Bare merely closed; o A and Bare open and bounded.

Let A, B C R. We define the distance between A and B by d(A, B) = inf{|x - y| : x E A, y E B}. Show that if A is compact and B is closed, and d(A, B) = 0, then ANB#Ø. • Give examples of d(A, B) = 0, ANB=Ø when o A and Bare merely closed; o A and Bare open and bounded.

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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Let A, B C R. We define the distance between A and B by d(A, B) = inf{|x - y| : x E A, y E B}. Show that if A is compact and B is closed, and d(A, B) = 0, then ANB#Ø. • Give examples of d(A, B) = 0, ANB=Ø when o A and Bare merely closed; o A and Bare open and bounded.
Transcribed Image Text:Let A, B C R. We define the distance between A and B by d(A, B) = inf{|x - y| : x E A, y E B}. Show that if A is compact and B is closed, and d(A, B) = 0, then ANB#Ø. • Give examples of d(A, B) = 0, ANB=Ø when o A and Bare merely closed; o A and Bare open and bounded.
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In mathematics, the infimum (abbreviated inf; plural infima) of a subset  of a partially ordered ser P is a greatest element in P that is less than or equal to all elements of S, if such an element exists.

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