Sometimes we state mathematical properties on a "simply connected open set". Some examples include Green Theorem, Cauchy Theorem, etc. Aside from being technically required by the proofs, what would be the intuition behind this condition, i.e. when will mathematicians resort to impose such a condition? What are the motivations?
2026-04-18 07:39:54.1776497994
Intuition behind Simply Connected Open Set
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Let $D\subset\mathbb C$ and let $F$ be a function from $D$ to $\mathbb C$. Generally, the need for the fact that $D$ is simply connected is because, when we have a closed loop $\gamma\colon[a,b]\longrightarrow\mathbb C$, it is needed for the proof of the theorem that the region of $\mathbb C$ enclosed by the image of $\gamma$ is part of the domain if $F$.