Topological argument principle

[Ashley1209]

TL;DR Summary

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?

 

[WWGD]

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?

 

[Ashley1209]

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.

 

[FactChecker]

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.

 

[mathwonk]

do you know about covering spaces?

 

[lavinia]

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.

 

[Ashley1209]

do you know about covering spaces?

Yes.Does this have something to do with covering spaces?

 

[lavinia]

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.

 

[mathwonk]

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".