# Prove that all contractible spaces are simply connected. [duplicate]

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, Daniel Rust, John MaMay 8 '14 at 9:48

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 Khan Feb 25 '13 at 20:34