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.

Prove that all contractible spaces are simply connected.

It's simple to prove that the space is pathwise connected. But, how can I prove that the fundamental group is trivial?

share|improve this question
add comment

2 Answers

up vote 5 down vote accepted

The fundamental group is homotopy invariant.

share|improve this answer
    
How can I prove that the fundamental group is trivial? –  Henfe Feb 25 '13 at 20:09
    
Do you know the definition of a contractible space and what homotopy invariance means? –  Adeel Feb 25 '13 at 20:13
    
Contractible spaces yes, I know. But, homotopy invariance not yet. I'm a student of math and I am studying algebraic topology. I starded to study it one month ago. –  Henfe Feb 25 '13 at 20:16
1  
Take a loop $f:[0,1] \to X$. Contractibility of $X$ is equivalent to every map $Y \to X$ being null-homotopic so in particular $f$ is null-homotopic. So the fundamental group consists of the single element which is the class of a constant map. –  Adeel Feb 25 '13 at 20:34
    
@Adeel Thank's! –  Nicole Feb 25 '13 at 20:38
add comment

Any closed curve can be contracted like the space.

share|improve this answer
add comment

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.