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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?

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marked as duplicate by ronno, Grigory M, mau, Dan Rust, John Ma May 8 '14 at 9:48

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2 Answers 2

up vote 5 down vote accepted

The fundamental group is homotopy invariant.

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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
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

Any closed curve can be contracted like the space.

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