Is a Locally One-to-One Proper Map Globally Bijective?

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Discussion Overview

The discussion revolves around the properties of a continuous, proper, and locally one-to-one map, specifically questioning whether such a map can be considered globally one-to-one. The scope includes theoretical aspects of topology, particularly concerning proper maps and covering spaces.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested

Main Points Raised

  • One participant asks if a continuous, proper, and locally one-to-one map is globally one-to-one.
  • Several participants seek clarification on the definition of a proper map, noting its relation to the inverse image of compact sets being compact.
  • Another participant suggests that the problem resembles a school textbook problem, implying a need for structured work rather than direct answers.
  • There is a mention of covering spaces, with one participant asking if the discussion relates to this concept.
  • A participant provides an analogy involving rubber bands to illustrate the concept of locally one-to-one mappings and their potential intersections.
  • Another participant discusses the properties of covering spaces, stating that all coverings are continuous and locally one-to-one, and that a compact covering space implies a proper covering map.
  • One participant reflects on the visualization of a specific mapping in the complex plane and connects it to the properties of continuous maps between nice spaces, suggesting that a locally bijective proper continuous map may be a finite covering of its image.

Areas of Agreement / Disagreement

Participants express varying levels of understanding regarding proper maps and covering spaces, with some agreeing on definitions while others raise questions or propose different perspectives. The discussion remains unresolved regarding whether a locally one-to-one proper map is globally bijective.

Contextual Notes

There are limitations in the discussion regarding the assumptions made about the nature of the spaces involved and the definitions of terms like "proper" and "one-to-one." The relationship between local and global properties in this context is not fully explored.

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φ is a continuous, proper and locally one-to-one map.Is it a globally one-to-one map?
φ is a continuous, proper and locally one-to-one map.Is it a globally one-to-one map?
 
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Welcome to PF.
Please remind us of what a proper map is. IIRC, it had something to see with inverse image of compact sets being compact?
 
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WWGD said:
Welcome to PF.
Please remind us of what a proper map is. IIRC, it had something to see with inverse image of compact sets being compact?
Yes, inverse image of compact sets being compact.And the map is between two topological discs.
 
This looks like a school textbook problem. For those, you must show work in a certain format. We are not supposed to give more than hints on your work.
 
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do you know about covering spaces?
 
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1) Take a rubber band and twist it into a figure 8.

2) Take a rubber band and push two oppose points together to make a figure 8.

Keep pushing so that the rubber band intersects itself in two points.

Try the same idea with a sphere.
 
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mathwonk said:
do you know about covering spaces?
Yes.Does this have something to do with covering spaces?
 
Ashley1209 said:
Yes.Does this have something to do with covering spaces?
All coverings are continuous and locally 1-1. If the covering space is compact then the covering map is also proper.

For instance, take a finite discrete set and map it onto one of its points.
 
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re: Lavinia's post #6, 1), can you visualize z --> z^2, for complex z: |z| = 1?

and I guess a continuous map between "nice" spaces (locally compact and Hausdorff?) should be a finite covering map if and only if it is a local homeomorphism, surjective and proper.

In fact since a continuous bijection of compact hausforff spaces is a homeomorphism, maybe even a locally bijective proper continuous map of locally compact hausdorff spaces is a finite covering of its image. So if "one to one" means "bijective", as it sometimes does, then this is why I was thinking of covering spaces as soon as I heard proper, continuous and locally one to one. I.e. that is essentially equivalent to "finite covering".
 
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