In topology: homeomorphism v. monotone function

Click For Summary

Homework Help Overview

The discussion revolves around the relationship between homeomorphisms and monotone functions in the context of topology. The original poster is exploring the conditions under which a bijection between real numbers can be classified as a homeomorphism based on its monotonicity.

Discussion Character

  • Exploratory, Assumption checking, Conceptual clarification

Approaches and Questions Raised

  • The original poster attempts to prove that a homeomorphism must be monotone by using a proof by contradiction. They raise questions about the implications of the definitions of homeomorphisms and monotone functions.
  • Some participants suggest using the intermediate value theorem and sketching to aid in understanding the problem.
  • Another participant questions whether the lack of monotonicity implies that the function takes the same value at different points.

Discussion Status

The discussion is active, with participants providing hints and suggestions that seem to help the original poster progress in their understanding. There is no explicit consensus, but guidance has been offered that appears to facilitate further exploration of the problem.

Contextual Notes

The original poster is working under the constraints of a homework assignment, which may limit the resources and methods they can use to arrive at a solution.

jjou
Messages
63
Reaction score
0
1. Let f:\mathbb{R}\rightarrow\mathbb{R} be a bijection. Prove that f is a homeomorphism iff f is a monotone function.



I think I have it one way (if f is monotone, it is a homeomorphism), but I'm stuck on the other way (if f is a homeomorphism, then it is monotone). I tried to prove this using contradiction.

Assume it is a homeomorphism but not monotone. Then there exists a,b,c in R s.t. a<b and a<c but f(a)<f(b) and f(a)>f(c). I think the statement I want to eventually contradict is the following: U\subset\mathbb{R} is open iff f(U)\subset\mathbb{R} is open.

Could someone give me a small hint? Thanks. :)
 
Physics news on Phys.org
Use the intermediate value theorem (draw a sketch!).
 
Can you prove that if f is not monotone, then f(x)= f(y) for some x\ne y?
 
Thanks! I got it very quickly using both of your suggestions. :D
 

Similar threads

Prove that f is a homeomorphism iff g is continuous, fg=1 and gf=1
  • · Replies 5 ·
Replies
5
Views
2K
Problem involving recursion theorem
  • · Replies 11 ·
Replies
11
Views
1K
Undergrad 2-sphere with any topology can't be homeomorphic to the plane
  • · Replies 5 ·
Replies
5
Views
2K
Spivak & Dimension of Manifold
  • · Replies 6 ·
Replies
6
Views
2K
Standard topology is coarser than lower limit topology?
  • · Replies 58 ·
2
Replies
58
Views
5K
Monotonic sequence definition of Continuity of a function
  • · Replies 13 ·
Replies
13
Views
2K
Prove every subset of countable set is either finite or else countable
  • · Replies 1 ·
Replies
1
Views
2K
Finding homeomorphism between topological spaces
  • · Replies 11 ·
Replies
11
Views
2K
Undergrad ##SU(2)## homeomorphic with ##\mathbb S^3##
  • · Replies 61 ·
3
Replies
61
Views
7K
Convergence of a continuous function related to a monotonic sequence
  • · Replies 3 ·
Replies
3
Views
2K