I am looking for a complex domain $D$ that is not simply connected, but the fact that $D$ is not simply connected is not obvious. For example, $\mathbb{C}\setminus \{p\}$ for some $p\in \mathbb{C}$ would be trivial because it's obvious. What I am trying to look for is a set which although it's not simply connected, it's not at all obvious that it's not simply connected. Another trivial domain would be a lemniscate, or the union of two disjoint simply connected domains.
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If we weaken from "domain" to "set" (= don't assume openness), then the Mandelbrot set counts: it's simply connected, but I don't think that's immediately obvious! See here.
Note that there are really two points going on here: that the Mandelbrot set is connected, and that the complement of the Mandelbrot set is connected. Together these imply simple-connectedness, and the latter by itself is called "fullness." For the Mandelbrot set, the latter is significantly easier to show than the former.
To get an actual domain, pick (say) the largest component of the interior of the Mandelbrot set. To me this is still non-obvious, it's just less cool. (It's easy to see that the interior of the Mandelbrot set itself is not connected.)
Noah Schweber
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Thank you, maybe I am mistaken, but I've read somewhere that the interior of the Mandelbrot set is simply connected. – Bertrand Wittgenstein's Ghost Jun 26 '23 at 05:21
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@BertrandWittgenstein'sGhost Can you find a citation for that? I think it's not too hard to show that the obvious vertical line separates the (interior of the) "main cardioid" from the (interior of the) "second-smallest bulb." – Noah Schweber Jun 26 '23 at 05:29
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Not sure if I can find one. Maybe I was mistaken. – Bertrand Wittgenstein's Ghost Jun 26 '23 at 06:02