On continuous and locally one-to-one map


by krete
Tags: continuous, locally, onetoone
krete
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#1
Jan31-13, 10:00 AM
P: 15
Whether a continuous and locally one-to-one map must be a (globally) one-to-one map? If the answer is not. Might you please give a counter-example? Thank in advance.
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jgens
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#2
Jan31-13, 10:35 AM
P: 1,623
The answer is no. Consider the mapping [itex]\mathbb{R} \rightarrow S^1[/itex] defined by [itex]x \mapsto \exp(2\pi i x)[/itex].
krete
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#3
Jan31-13, 11:03 AM
P: 15
Got it, many thanks!

Another question: whether a continuous and locally one-to-one map between two open spaces, e.g., two connected open set of R^n, must be a (globally) one-to-one map?

micromass
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#4
Jan31-13, 11:53 AM
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On continuous and locally one-to-one map


Quote Quote by krete View Post
Got it, many thanks!

Another question: whether a continuous and locally one-to-one map between two open spaces, e.g., two connected open set of R^n, must be a (globally) one-to-one map?
No, consider

[tex]\mathbb{C}\setminus \{0\}\rightarrow \mathbb{C}:z\rightarrow z^2[/tex]
krete
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#5
Jan31-13, 12:11 PM
P: 15
Thanks a lot!


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