In topology: homeomorphism v. monotone function

  • Thread starter jjou
  • Start date
  • #1
64
0
1. Let [tex]f:\mathbb{R}\rightarrow\mathbb{R}[/tex] 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: [tex]U\subset\mathbb{R}[/tex] is open iff [tex]f(U)\subset\mathbb{R}[/tex] is open.

Could someone give me a small hint? Thanks. :)
 

Answers and Replies

  • #2
morphism
Science Advisor
Homework Helper
2,015
4
Use the intermediate value theorem (draw a sketch!).
 
  • #3
HallsofIvy
Science Advisor
Homework Helper
41,847
964
Can you prove that if f is not monotone, then f(x)= f(y) for some [itex]x\ne y[/itex]?
 
  • #4
64
0
Thanks! I got it very quickly using both of your suggestions. :D
 

Related Threads on In topology: homeomorphism v. monotone function

  • Last Post
Replies
1
Views
2K
Replies
5
Views
4K
Replies
11
Views
695
Replies
3
Views
1K
Replies
2
Views
2K
Replies
1
Views
3K
  • Last Post
Replies
3
Views
3K
  • Last Post
Replies
15
Views
2K
  • Last Post
Replies
0
Views
1K
Replies
2
Views
3K
Top