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There is a ton of equivalent definitions of a simply connected set on the plane, reading Ullrich's Complex Made Simple I have some of these equivalences, my question is about (iii) and (iv) equivalences (where smooth means piecewise $C^1$ and simply connected means all curves are homotopic to a point):

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I'm having some trouble because of another result in Conway's Functions of One Complex Variable I. It says that exists a closed (rectifiable) curve with $\text{Ind}(\gamma,z)=0, z \notin G$, that is not homotopic to a point.

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My two thoughts were: Or the difference is cause of the more general curve, by being only rectifiable and not $C^1$, or the rectifiable doesn't matter here and these things are not a contradiction, because (iv) needs that every curve fullfil this condition, and not that this condition on one curve implies the curve being homotopic to a point. One of them make sense?

And by last it would be great if someone could tell me where can I find a proof of simply-connected if and only if every closed curve fullfil the winding number condition.

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