# Equivalence between definitions of simple connectedness.

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.

• @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. – Math1000 Oct 30 at 22:10
• 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. – Math1000 Nov 11 at 2:01
• Once one proves the above problem, only then one can start developing an intuition about holes -- not vice versa. – Wlod AA Nov 11 at 3:05

The intuition to have here, is to think that $$X$$ has no holes in it. Both definitions make perfect sense from this point on.