On continuous and locally one-to-one map

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Discussion Overview

The discussion centers on whether a continuous and locally one-to-one map must also be a globally one-to-one map. Participants explore this question through examples and counterexamples, particularly in the context of mappings between open sets in \(\mathbb{R}^n\) and \(\mathbb{C}\).

Discussion Character

  • Debate/contested

Main Points Raised

  • One participant questions if a continuous and locally one-to-one map must be globally one-to-one, seeking a counterexample.
  • Another participant provides a counterexample using the mapping from \(\mathbb{R}\) to \(S^1\) defined by \(x \mapsto \exp(2\pi i x)\), asserting that it is not globally one-to-one.
  • A subsequent participant reiterates the question regarding continuous and locally one-to-one maps between two connected open sets in \(\mathbb{R}^n\), seeking clarification.
  • Another participant responds with a counterexample involving the mapping from \(\mathbb{C} \setminus \{0\}\) to \(\mathbb{C}\) defined by \(z \mapsto z^2\), indicating it is also not globally one-to-one.

Areas of Agreement / Disagreement

Participants do not reach a consensus, as multiple counterexamples are presented, suggesting that the initial question remains unresolved.

Contextual Notes

The discussion highlights the need for careful consideration of the definitions of continuity and local one-to-one mappings, as well as the implications of these properties in different contexts.

krete
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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?
 
krete said:
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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