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Math Help - Contractible space

  1. #1
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    Contractible space

    I' working on this problem, and I got stuck in the middle. Could someone give me a hand? Let X be a contractible space. I'd like to show that X is simply connected. I already showed that X is path connected. I need to show for every x_0 \in  X, \pi_1(X,x_0) is the trivial group. So, it suffices to show every loop based at x_0 is path homotopic to the path e: I\to X defined by e(x)=x_0 for every x\in X

    Let \alpha be a loop based at fixed x_0 \X. X is contractible implies there exists a homotopy H: X \times I \to X such that H(x,0)=x and H(x,1)=x_0. I tried to construct a path homotopy from \alpha to e using H, but I don't think mine works. I define G(x,t)= H(\alpha(x),t). Then G(x,0)=H(\alpha(x),0)=\alpha(x) and G(x,1)=H(\alpha(x),1)=x_0. But this is only a homotopy, I want H(0,t)=H(1,t)=x_0. I can't get it with this map.
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