
#1
Jan3113, 10:00 AM

P: 15

Whether a continuous and locally onetoone map must be a (globally) onetoone map? If the answer is not. Might you please give a counterexample? Thank in advance.




#2
Jan3113, 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].




#3
Jan3113, 11:03 AM

P: 15

Got it, many thanks!
Another question: whether a continuous and locally onetoone map between two open spaces, e.g., two connected open set of R^n, must be a (globally) onetoone map? 



#4
Jan3113, 11:53 AM

Mentor
P: 16,545

On continuous and locally onetoone map[tex]\mathbb{C}\setminus \{0\}\rightarrow \mathbb{C}:z\rightarrow z^2[/tex] 



#5
Jan3113, 12:11 PM

P: 15

Thanks a lot!



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