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On continuous and locally one-to-one map

  1. Jan 31, 2013 #1
    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.
     
  2. jcsd
  3. Jan 31, 2013 #2

    jgens

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    Gold Member

    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].
     
  4. Jan 31, 2013 #3
    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?
     
  5. Jan 31, 2013 #4
    No, consider

    [tex]\mathbb{C}\setminus \{0\}\rightarrow \mathbb{C}:z\rightarrow z^2[/tex]
     
  6. Jan 31, 2013 #5
    Thanks a lot!
     
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