On continuous and locally one-to-one map

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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 The answer is no. Consider the mapping $\mathbb{R} \rightarrow S^1$ defined by $x \mapsto \exp(2\pi i x)$.
 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?

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On continuous and locally one-to-one map

 Quote by 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?
No, consider

$$\mathbb{C}\setminus \{0\}\rightarrow \mathbb{C}:z\rightarrow z^2$$
 Thanks a lot!

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