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Let A, B and X be topological spaces. If A can not be embedded into B then A × X can not be embedded into B x X. This conjecture turns out to be false however. We can take A = S¹ and B = X = R as a counterexample. (a) Proof that this is indeed a counterexample, i.e. show that S¹ can not be embedded into R, but S¹ × R can be embedded into R².

Let A, B and X be topological spaces. If A can not be embedded into B then A × X can not be embedded into B x X. This conjecture turns out to be false however. We can take A = S¹ and B = X = R as a counterexample. (a) Proof that this is indeed a counterexample, i.e. show that S¹ can not be embedded into R, but S¹ × R can be embedded into R².

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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Let A, B and X be topological spaces. If A can not be embedded into B then A × X can not be embedded into B × X. This conjecture turns out to be false however. We can take A = S¹ and B = X = R as a counterexample. (a) Proof that this is indeed a counterexample, i.e. show that S¹ can not be embedded into R, but S¹ × R can be embedded into R².
Transcribed Image Text:Let A, B and X be topological spaces. If A can not be embedded into B then A × X can not be embedded into B × X. This conjecture turns out to be false however. We can take A = S¹ and B = X = R as a counterexample. (a) Proof that this is indeed a counterexample, i.e. show that S¹ can not be embedded into R, but S¹ × R can be embedded into R².
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