2.4.7. Let A be a subset of a metric space X. Given x E X, define the distance from x to A to be dist(x, A) = inf{d(x, y) : y E A}. 2.4 Closed Sets 61 Prove the following statements. (a) If A is closed, then x E A if and only if dist(x, A) = 0. (b) dist(x, A) < d(x, y) + dist(y, A) for all x, y e X. (c) |dist(x, A) – dist(y, A)| < d(x, y) for all æ, y E X. Additionally, show by example that it is possible to have dist(x, A) = 0 even when x ¢ A.

2.4.7. Let A be a subset of a metric space X. Given x E X, define the distance from x to A to be dist(x, A) = inf{d(x, y) : y E A}. 2.4 Closed Sets 61 Prove the following statements. (a) If A is closed, then x E A if and only if dist(x, A) = 0. (b) dist(x, A) < d(x, y) + dist(y, A) for all x, y e X. (c) |dist(x, A) – dist(y, A)| < d(x, y) for all æ, y E X. Additionally, show by example that it is possible to have dist(x, A) = 0 even when x ¢ A.

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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