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Wikipedia gives two definitions of simple connectedness:

  1. A topological space $X$ is simply connected iff it is path-connected and any loop in $X$ can be contracted to a point.
  2. A topological space $X$ is simply connected iff it is path-connected and any two paths with the same start-point and end-point are homotopic.

I do not see how these are equivalent. What is the connection between loops being contractable and and paths being homotopic? I am mostly looking for intuition here, as opposed to a formal proof.

By the way, I am aware of another definition: A topological space $X$ is simply connected iff its fundamental group is trivial. I would like to avoid that characterization here.

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    $\begingroup$ @MoisheKohan This is not a duplicate. That question is referring to the characterization of simple connectedness by triviality of the fundamental group. Here I am specifically avoiding that. $\endgroup$ – Math1000 Oct 30 at 22:10
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    $\begingroup$ As I explained in my above comment, this question is NOT a duplicate of the other question. The other question uses trivial fundamental group as a definition of simple connectivity; here I am not. $\endgroup$ – Math1000 Nov 11 at 2:01
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    $\begingroup$ Once one proves the above problem, only then one can start developing an intuition about holes -- not vice versa. $\endgroup$ – Wlod AA Nov 11 at 3:05
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The intuition to have here, is to think that $X$ has no holes in it. Both definitions make perfect sense from this point on.

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    $\begingroup$ Indeed that is a great intuition to have for the definition itself, but I am looking more for intuition for the equivalence of definitions 1 and 2. $\endgroup$ – Math1000 Oct 29 at 21:39
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    $\begingroup$ If you want more mathematical insight into why they are indeed equivalent, take one loop and consider that the loop is a path and that the point at which it is contracted is also a path. Using the second definition you easily get the first one. $\endgroup$ – Theleb Oct 29 at 21:46
  • $\begingroup$ The 1) => 2) implication must certainly be less intuitive $\endgroup$ – Theleb Oct 29 at 21:48
  • $\begingroup$ I am guessing that taking any two paths, thanks to the connectedness of X you can construct a loop containing the two paths. Then using the first definition you can contract these two paths into a single point. Then you can (maybe with some efforts) construct a homotopy between the two paths. $\endgroup$ – Theleb Oct 29 at 22:05
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    $\begingroup$ Ah, I left out an important detail of the definition 2: we only consider paths with the same start-point and end-point. $\endgroup$ – Math1000 Oct 29 at 22:21

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