Ashley1209
- 3
- 0
- TL;DR
- 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?
Is a Locally One-to-One Proper Map Globally Bijective?
- Context: Undergrad
- Thread starter Ashley1209
- Start date
-
- Tags
- Continuous
Join the discussion
Registration is free. Ask a follow-up in this thread, or start your own.
8 replies · 3K views
Discussion
Physics news on Phys.org
- 7,830
- 13,158
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?
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:
Ashley1209
- 3
- 0
Yes, inverse image of compact sets being compact.And the map is between two topological discs.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?
- 9,551
- 4,790
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:
- 12,021
- 2,319
do you know about covering spaces?
- 3,385
- 760
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.
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:
Ashley1209
- 3
- 0
Yes.Does this have something to do with covering spaces?mathwonk said:do you know about covering spaces?
- 3,385
- 760
All coverings are continuous and locally 1-1. If the covering space is compact then the covering map is also proper.Ashley1209 said:Yes.Does this have something to do with covering spaces?
For instance, take a finite discrete set and map it onto one of its points.
Last edited:
- 12,021
- 2,319
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".
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:
Similar threads
Graduate On continuous and locally one-to-one map
- krete
- · Replies 4 ·
- Topology and Analysis
- Replies
- 4