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

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In summary, the article explores the relationship between locally one-to-one proper maps and their potential to be globally bijective. It examines the conditions under which a map that is locally one-to-one and proper can be guaranteed to be globally bijective, discussing various mathematical frameworks and examples that illustrate these concepts. The findings highlight the importance of additional topological and geometrical properties in determining global behavior from local characteristics.
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Ashley1209
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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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  • #2
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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  • #3
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.
 
  • #4
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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  • #5
do you know about covering spaces?
 
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  • #6
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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  • #7
mathwonk said:
do you know about covering spaces?
Yes.Does this have something to do with covering spaces?
 
  • #8
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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FAQ: Is a Locally One-to-One Proper Map Globally Bijective?

1. What is a locally one-to-one proper map?

A locally one-to-one proper map is a function between two topological spaces that is locally injective (meaning that around every point in the domain, there exists a neighborhood such that the map restricted to that neighborhood is injective) and is a proper map (meaning the preimage of every compact set is compact). This type of map preserves local structure and is well-behaved with respect to compactness.

2. What does it mean for a map to be globally bijective?

A map is globally bijective if it is both injective (one-to-one) and surjective (onto) across the entire domain and codomain. This means that for every point in the codomain, there is exactly one point in the domain that maps to it, establishing a one-to-one correspondence between the two spaces.

3. Can a locally one-to-one proper map fail to be globally bijective?

Yes, a locally one-to-one proper map can fail to be globally bijective. While local injectivity ensures that small neighborhoods around each point do not overlap, it does not guarantee that the entire map is injective or that it covers the entire codomain. There may be points in the codomain that are not mapped to by any point in the domain, or the map may not be injective on a global scale.

4. Under what conditions can a locally one-to-one proper map be guaranteed to be globally bijective?

A locally one-to-one proper map can be guaranteed to be globally bijective if the domain is compact and the codomain is Hausdorff. In this scenario, the properness of the map ensures that it is closed and that the image of the compact space is compact, which, combined with the properties of Hausdorff spaces, implies that the map is both injective and surjective.

5. What are some examples of locally one-to-one proper maps that are not globally bijective?

An example of a locally one-to-one proper map that is not globally bijective is the map from the open interval (0, 1) to the real line R defined by f(x) = tan(π(x - 0.5)). This map is locally one-to-one because it is injective on small intervals, and it is proper because the preimage of compact sets in R will be compact in (0, 1). However, it is not globally bijective since it does not cover all of R and has vertical asymptotes, leading to points in R that do not correspond to any point in (0, 1).

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