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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 2.38. Show by example(s) that Proposition 2.37 does not generally hold if Y is not closed (respectively, not open) in X.

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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solve 2.38

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 (respectively, not open) in X.
Transcribed Image Text: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 (respectively, not open) in X.
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Step 1

Let the space be X=ℝ with usual topology and let Y=[1,2) is a subspace of ℝ but it is neither open nor closed under usual topology. 

Let Y=[1,2) has the subspace topology i.e any open set in Y is of the form U∩[1,2) where U is open in the space X with usual topology.

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