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Prove that a star-shaped open set is simply connected.

This is an exercise in Stein's complex analysis and I have already seen one of the proof here.
In Stein's book , A region $\Omega$ in the complex plane is simply connected if any two pair of curves in $\Omega$ with the same end-points are homotopic. But in the proof above , it seems to apply another definition of simply connected (any closed curve in the region is homotopic to a point in the region?) .
My question:
a) How to prove this exercise with the definition of simple connected in Stein's book .
b) What kind of the definition of simple connected did the proof above applied . Are the two definition of simple connected the same ? If so , how to show this ?

J.Guo
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  • Hint: We can deform any curve into the curve defined by straight lines from the end points to the center point. To do this, start by taking an open ball around one of your endpoints and intersecting it with your starting curve. This section of your starting curve is contained in a convex subset of $\Omega$. Moreover, it is contained in a convex subset containing the line from your endpoint to the center point, and can therefore be deformed into that line continuously. Do this with both endpoints, and send all other points to the center point to get a complete homotopy. – Miles Johnson Oct 22 '20 at 06:48

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