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
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
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Question
Solve 2.36 and 2.37

Transcribed Image Text: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.
Proposition 2.37. Let X be a space, Y c X, and z CY. 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 (respреctively, not opеn) in X.
Problem 2.39. Let X be a space and Y c X. Giwe Y the subspace topology. Let A CY.
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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