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Here, page $458$, in the proof of theorem $2$, there is this sentence

'Conversely, suppose $G=\bigcup_{i \geq 1}{U_i}$ where $U_i$ are open in $[0,1] \times [0,1]$ and simply connected. For each positive integer $n$, let $V_n=\bigcap_{i=1}^n{U_i}$ containing the connected $G$. It is easy to verify that each $V_n$ is simply connected.'

I don't understand why $V_n$ is simply connected. Can someone enlighten me?

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  • $\begingroup$ The intersection of 2 "fighting" crescents is not simply connected although each of them is. $\endgroup$ – Adelafif Aug 13 '15 at 10:42
  • $\begingroup$ there is a misprint, $G=\bigcap_{i \geq 1}{U_i}$. $\endgroup$ – Andrey Ryabichev Aug 13 '15 at 17:59
  • $\begingroup$ and one more: $V_n$ is a component of $\bigcap_{i=1}^nU_i$ containing $G$. so it is an answer for your question. $\endgroup$ – Andrey Ryabichev Aug 13 '15 at 18:02

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