Topological argument principle

  • I
  • Thread starter Ashley1209
  • Start date
  • Tags
    Continuous
  • #1
Ashley1209
3
0
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?
 
Physics news on Phys.org
  • #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?
 
Last edited:
  • #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.
 
Last edited:
  • #5
do you know about covering spaces?
 
  • Like
Likes lavinia and WWGD
  • #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.
 
Last edited:
  • #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.
 
Last edited:
  • Like
Likes Ashley1209
  • #9
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".
 
Last edited:
  • Like
Likes Ashley1209 and lavinia

Similar threads

  • Topology and Analysis
2
Replies
43
Views
780
  • Topology and Analysis
Replies
8
Views
1K
  • Topology and Analysis
Replies
8
Views
378
Replies
15
Views
1K
  • Topology and Analysis
Replies
5
Views
1K
  • Topology and Analysis
Replies
8
Views
1K
Replies
6
Views
2K
  • Topology and Analysis
Replies
2
Views
2K
  • Topology and Analysis
Replies
5
Views
2K
  • Topology and Analysis
Replies
5
Views
1K
Back
Top