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3. For a metric space (X, d), the diameter of a subset A C X is defined by 6(A) := sup d(x, y). x,yƐA Here we allow 8(A) = o. Show that if A, B C X satisfy AN B + 0, then 8(AUB) < 8(A) + 8(B). Give an example to show this is not necessarily true when ANB = Ø.

3. For a metric space (X, d), the diameter of a subset A C X is defined by 6(A) := sup d(x, y). x,yƐA Here we allow 8(A) = o. Show that if A, B C X satisfy AN B + 0, then 8(AUB) < 8(A) + 8(B). Give an example to show this is not necessarily true when ANB = Ø.

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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Metric spaces

3. For a metric space (X, d), the diameter of a subset AC X is defined by 8(A) := sup d(x, y). x,yƐA Here we allow 8(A) = ∞. Show that if A, B C X satisfy ANB + Ø, then 8(AU B) < 8(A) + 8(B). Give an example to show this is not necessarily true when AN B = Ø.
Transcribed Image Text:3. For a metric space (X, d), the diameter of a subset AC X is defined by 8(A) := sup d(x, y). x,yƐA Here we allow 8(A) = ∞. Show that if A, B C X satisfy ANB + Ø, then 8(AU B) < 8(A) + 8(B). Give an example to show this is not necessarily true when AN B = Ø.
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