Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

I need to solve this problem, but I don't know how start.

Let be $(X,\tau)$ a topological space, and $G$ the set of all homeomorphism of $X$. I just proved that $G$ is a group, but I need also prove this

  1. If $X = [0,1]$, then $G$ is infinite.
  2. If $X = [0,1]$, is $G$ abelian?

I appreciate all your comments. Thank you.

share|improve this question

2 Answers 2

up vote 5 down vote accepted

I would suggest the following hints:

  1. Consider the maps $x\to x^n$ for $x\in [0,1]$ and $n\in \mathbb{N}$.

  2. Consider the map $x\to 1-x$ for $x\in [0,1]$.

Exercise 1: Prove that there exists a homeomorphism of $[0,1]$ of infinite order. (Note that this is stronger than 1.)

Exercise 2: Does there exist a homeomorphism of $[0,1]$ of order $k$ for each $k\in \mathbb{N}$?

Exercise 3: Prove that $G$ is uncountable.

Exercise 4: Does there exist a subgroup of $G$ isomorphic to $(\mathbb{R}, +)$?

Exercise 5: Can you exhibit a non-trivial proper normal subgroup of $G$?

Exercise 6: Prove that $G$ does not act transitively on $[0,1]$. What is the orbit of $\{x\}$ under the action of $G$ for $x\in [0,1]$?

In general, I recommend trying to come up with as many exercises as you can about $G$ in order to obtain a better understanding.

Hope this helps!

share|improve this answer
    
(+1) indeed :) :) –  Ittay Weiss Apr 23 at 0:48

1) Start by constructing some homeomorphisms of the $[0,1]$. Hint: $x^2, x^3, x^4,\cdots $

2) Are there some homeomorphisms that reflect things?

share|improve this answer
1  
We have almost identical answers so +1 :) –  Amitesh Datta Apr 23 at 0:24

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.