Topological argument principle

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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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What is the Topological argument principle?

The Topological argument principle is a mathematical theorem that states that if a function is continuous on a closed interval and changes sign at the endpoints of the interval, then there exists at least one point within the interval where the function is equal to zero.

What is the significance of the Topological argument principle?

The Topological argument principle is significant because it provides a method for proving the existence of solutions to certain types of equations. It is commonly used in analysis and calculus to show the existence of roots or zeros of functions.

How is the Topological argument principle used in mathematics?

In mathematics, the Topological argument principle is used to prove the existence of solutions to equations by showing that a function changes sign within a closed interval. This allows mathematicians to find points where the function is equal to zero, which can be useful in solving various problems.

Are there any limitations to the Topological argument principle?

One limitation of the Topological argument principle is that it only guarantees the existence of at least one solution within a given interval. It does not provide information about the uniqueness of the solution or how to find it. Additionally, the principle may not be applicable to all types of functions or equations.

Can the Topological argument principle be applied to real-world problems?

Yes, the Topological argument principle can be applied to real-world problems in various fields such as physics, engineering, and economics. By proving the existence of solutions to equations, this principle can help in analyzing and solving practical problems that involve continuous functions and intervals.

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