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Let $D$ be a convex domain in the complex plane, and is domain $D$ a simply connected domain? What about when $D$ is a star domain?

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Yes star-domains are simply connected, since every path is homotopy equivalent to one going through the center.

The disc with one point removed is not simply connected, but also not convex.

Open convex sets are among the star-domains.

All that is not special to the $\mathbb{C}$, but any $\mathbb{R}^d$.

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the disc with one point removed is not a convex domain! – Riemann Apr 15 '12 at 13:09
The disc with one (interior) point removed is not convex, either, right? – Gerry Myerson Apr 15 '12 at 13:09
You can check using the definition do convex domain!!! – Riemann Apr 15 '12 at 13:12
Okay I missed the convex assumption;( Yes convex open sets are connected and simply connected. – Apr 15 '12 at 13:13
@late_learner can you give me a proof? thank you! – Riemann Apr 15 '12 at 13:15

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