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 $\mathbb{R} \rightarrow S^1$ defined by $x \mapsto \exp(2\pi i x)$.

 

[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]

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

$$\mathbb{C}\setminus \{0\}\rightarrow \mathbb{C}:z\rightarrow z^2$$

 

[krete]

Thanks a lot!