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  3. 2 let xd be metric space and let scycx a...

Question: 2 let xd be metric space and let scycx a...

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2. Let (X,d) be metric space, and let SCYcX. (a) Prove that S is an open subset of the metric space (Y, d) if and only if there exists an open subset U of the metric space (X, d) such that S = Un Y. (Hint: First consider y e Y and open balls centered at y.) (b) Deduce that S is closed relative to Y if and only if there exists a closed subset F of (X,d) such that S = FnY (c) Prove that S is compact as a subset of (Y, d) if and only if it is compact as a subset of (X, d). (d) Deduce that Y is a compact subset of (X, d) if and only if the metric space (Y,d) is compact.

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