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a) Suppose that f is a surjective local homeomorphism. Show that the topology Ty of Y equals the set S={f(U) |UE Tx}. It can be shown that any covering map is a local homeomorphism. Recall that p: R→ S¹ defined by p(t) = (cos(2nt), sin(2nt)) is a covering map. → b) Give an open subset X CR such that f: X S¹, given by f(t) = (cos(2nt), sin(2t)) is surjective, a local homeomorphism but not a covering map. Explain your answer.

a) Suppose that f is a surjective local homeomorphism. Show that the topology Ty of Y equals the set S={f(U) |UE Tx}. It can be shown that any covering map is a local homeomorphism. Recall that p: R→ S¹ defined by p(t) = (cos(2nt), sin(2nt)) is a covering map. → b) Give an open subset X CR such that f: X S¹, given by f(t) = (cos(2nt), sin(2t)) is surjective, a local homeomorphism but not a covering map. Explain your answer.

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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Question 4. Let X, Y be topological spaces. A continuous map f: X→ Y is a local homeomorphism if for every x EX there exists a neighborhood UC X of z, such that f(U) is open in Y and J: U → f(U) defined by f(x) = f(x) is a homeomorphism. Let Tx be the topology on X and Ty the topology on Y. a) Suppose that f is a surjective local homeomorphism. Show that the topology Ty of Y equals the set S={f(U) |UE Tx}. It can be shown that any covering map is a local homeomorphism. Recall that p: R → S¹ defined by p(t) = (cos(2nt), sin(2nt)) is a covering map. b) Give an open subset X CR such that f: X S¹, given by f(t) = (cos(2nt), sin(2t)) is surjective, a local homeomorphism but not a covering map. Explain your answer.
Transcribed Image Text:Question 4. Let X, Y be topological spaces. A continuous map f: X→ Y is a local homeomorphism if for every x EX there exists a neighborhood UC X of z, such that f(U) is open in Y and J: U → f(U) defined by f(x) = f(x) is a homeomorphism. Let Tx be the topology on X and Ty the topology on Y. a) Suppose that f is a surjective local homeomorphism. Show that the topology Ty of Y equals the set S={f(U) |UE Tx}. It can be shown that any covering map is a local homeomorphism. Recall that p: R → S¹ defined by p(t) = (cos(2nt), sin(2nt)) is a covering map. b) Give an open subset X CR such that f: X S¹, given by f(t) = (cos(2nt), sin(2t)) is surjective, a local homeomorphism but not a covering map. Explain your answer.
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