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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).

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).

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
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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).
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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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).
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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