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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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
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Do you know the definition of a contractible space and what homotopy invariance means? – Feb 25 '13 at 20:13
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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
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1Take 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. – Feb 25 '13 at 20:34
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@Adeel Thank's! – Nicole Feb 25 '13 at 20:38
