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Problem 2.36. Let X be a space and Y C X. Give Y the subspace topology. Describe (in a useful way) the closed sets in Y.

Problem 2.36. Let X be a space and Y C X. Give Y the subspace topology. Describe (in a useful way) the closed sets in Y.

Advanced Engineering Mathematics
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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Please solve 2.36 and 2.37

Problem 2.36. Let X be a space and YC X. Give Y the subspace topology. Describe (in a useful way) the closed sets in Y. Proposition 2.37. Let X be a space, Y C X, and Z C Y. Give Y the subspace topology. The following hold: (1) Suppose Y is closed in X. Then Z closed in Y implies Z closed in X. (2) Suppose Y is open in X. Then Z open in Y implies Z open in X. Problem 2.38. Show by example(s) that Proposition 2.37 does not generally hold if Y is not closed (respectively, not open) in X. Problem 2.39. Let X be a space and Y C X. Give Y the subspace topology. Let ACY. The notation A is ambiguous. Do we mean the closure of A in the original space X or the closure of A in the subspace Y? Show by example that this is really ambiguous. Second point: Find a good disambiguating notational solution.
Transcribed Image Text:Problem 2.36. Let X be a space and YC X. Give Y the subspace topology. Describe (in a useful way) the closed sets in Y. Proposition 2.37. Let X be a space, Y C X, and Z C Y. Give Y the subspace topology. The following hold: (1) Suppose Y is closed in X. Then Z closed in Y implies Z closed in X. (2) Suppose Y is open in X. Then Z open in Y implies Z open in X. Problem 2.38. Show by example(s) that Proposition 2.37 does not generally hold if Y is not closed (respectively, not open) in X. Problem 2.39. Let X be a space and Y C X. Give Y the subspace topology. Let ACY. The notation A is ambiguous. Do we mean the closure of A in the original space X or the closure of A in the subspace Y? Show by example that this is really ambiguous. Second point: Find a good disambiguating notational solution.
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