On continuous and locally one-to-one map

krete

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.

jgens

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

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

No, consider

[tex]\mathbb{C}\setminus \{0\}\rightarrow \mathbb{C}:z\rightarrow z^2[/tex]

krete

Thanks a lot!