6. Let A and B be two connected subsets of X and An B + 4. Show that AUB is connected. In R*(eqquiped with the Euclidean metric), find if the union of A = {(x,y): x² + y s 1} and B = ((x,y): (x - 2)2 + y < 1} is connected or not? Justify. Let (X, d) be a compact metric space and let d, be the metric on X defined by d, (x, y) = min{ 1, d(x, y)}. x, y EX Then prove that (X, d,) is compact.
6. Let A and B be two connected subsets of X and An B + 4. Show that AUB is connected. In R*(eqquiped with the Euclidean metric), find if the union of A = {(x,y): x² + y s 1} and B = ((x,y): (x - 2)2 + y < 1} is connected or not? Justify. Let (X, d) be a compact metric space and let d, be the metric on X defined by d, (x, y) = min{ 1, d(x, y)}. x, y EX Then prove that (X, d,) is compact.
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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Transcribed Image Text:6. Let A and B be two connected subsets of X and AnB + 4. Show that AUB is connected. In
R*(eqquiped with the Euclidean metric), find if the union of A = {(x, y): x* + y? s 1} and
B ((x,y): (x- 2)2 + y² < 1} is connected or not? Justify.
Let (X, d) be a compact metric space and let d, be the metric on X defined by
d, (x, y) = min{ 1, d(x, y)},
x, y EX
Then prove that (X, d,) is compact.
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