Let (X, d) be a metric space, and Y be a non-empty subset of X. (i) Equip Y with the distance defined by restricting d to Y × Y, which we denote by d again. Prove that (Y, d) is a metric space as well. Notation: We say (Y, d) is a metric subspace of (X, d). (ii) Suppose SY ≤ X. Prove that S is compact in (X, d) if and only if S is compact in the metric subspace (Y, d).
Let (X, d) be a metric space, and Y be a non-empty subset of X. (i) Equip Y with the distance defined by restricting d to Y × Y, which we denote by d again. Prove that (Y, d) is a metric space as well. Notation: We say (Y, d) is a metric subspace of (X, d). (ii) Suppose SY ≤ X. Prove that S is compact in (X, d) if and only if S is compact in the metric subspace (Y, d).
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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(ii) please!
![Let (X, d) be a metric space, and Y be a non-empty subset of X.
(i) Equip Y with the distance defined by restricting d to Y × Y, which we
denote by d again. Prove that (Y, d) is a metric space as well.
Notation: We say (Y, d) is a metric subspace of (X, d).
(ii) Suppose SC Y ≤ X. Prove that S is compact in (X, d) if and only if S is
compact in the metric subspace (Y, d).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe15ed467-90ec-4e60-afef-3d3f6119f74d%2F21607d25-7f6f-4d3f-8868-7d9ae7658dbf%2Fgd0of7c_processed.png&w=3840&q=75)
Transcribed Image Text:Let (X, d) be a metric space, and Y be a non-empty subset of X.
(i) Equip Y with the distance defined by restricting d to Y × Y, which we
denote by d again. Prove that (Y, d) is a metric space as well.
Notation: We say (Y, d) is a metric subspace of (X, d).
(ii) Suppose SC Y ≤ X. Prove that S is compact in (X, d) if and only if S is
compact in the metric subspace (Y, d).
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