Simple Topology Question

  • Thread starter jimisrv
  • Start date
7
0
Hi,

I have a question that I'm not sure about.
If f:A->C is continuous and B is a subset of C that is simply connected, is f(^-1)(B) necessarily connected or simply connected for that matter? Since the spaces are not necessarily homeomorphic I cannot consider it a topological invariant.

Thanks
 

quasar987

Science Advisor
Homework Helper
Gold Member
4,771
6
Consider the old canonical mapping from [0,2pi) to the circle S^1 in the complex plane (f(t)=e^it). Then for r small enough, f^-1(B(1,r)) is clearly not connected (and hence not simply connected either) since it consist of a little neighborhoods of 0 and a little neighborhoods of 2pi.
 
255
0
Actually, we can be even simpler than that: let f:R->R be given by f(x)=x^2, and let B={1}. Then f^-1(B)={1,-1}.
 
7
0
Thanks for the help! The last one is a very simple counterexample.
 

WWGD

Science Advisor
Gold Member
4,304
1,867
Maybe for a more extreme counterexample re connectedness, consider the case of

IR as a covering space of S^1 . Maybe if you had 1-1 -ness. (tho not in this case,

since continuous bijection bet. compact and hausdorff is a homeo., which is sufficient,

tho I don't know if it is necessary).


For a trivial counterexample re simple-connectedness, consider a constant map

defined on an annulus.
 

mathwonk

Science Advisor
Homework Helper
10,720
894
trivial counterexamples do exist for zero dimensional B, but try higher dimensional B's.

and try it for algebraic maps.

i.e. if you map an algebraic variety X to an algebraic variety Y, what does the inverse image of an irreducible curve in Y look like?

more specifically, project a surface onto P^2, and ask what the inverse image of a general line looks like?

see the fulton - hansen connectedness theorem, and various versions of bertini's thoerem.
 
Last edited:

mathwonk

Science Advisor
Homework Helper
10,720
894
here is a nice theorem of fulton and collaborators:

if L is a linear subspace of a projective space P, having codimension d,

and if X-->P is any morphism from a projective variety X,

having image in P of dimension larger than d,

then the inverse image in X of L is connected.
 

Related Threads for: Simple Topology Question

  • Posted
Replies
12
Views
2K
  • Posted
Replies
3
Views
4K
  • Posted
Replies
6
Views
3K
  • Posted
Replies
2
Views
2K
  • Posted
Replies
1
Views
2K
  • Posted
2
Replies
27
Views
5K
  • Posted
2
Replies
33
Views
7K
Replies
3
Views
2K

Physics Forums Values

We Value Quality
• Topics based on mainstream science
• Proper English grammar and spelling
We Value Civility
• Positive and compassionate attitudes
• Patience while debating
We Value Productivity
• Disciplined to remain on-topic
• Recognition of own weaknesses
• Solo and co-op problem solving
Top