find the example that $\bigcap\limits _n A_n$ is not connected?

Asked Dec 19 '19 at 19:57

Active Dec 19 '19 at 19:59

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Give an example of a sequence $(A_n)$ of connected subset of $\mathbb{R}^2$ such that $A_{n+1} \subset A_n$ for $n \in \mathbb{N} $ but $\bigcap\limits_n A_n$ is not connected

My attempt : I take $A_n=(-\frac{1}{n}-1,\frac{1}{n}+1)$ but $\bigcap\limits_{n=1}^{\infty} (-\frac{1}{n}-1,\frac{1}{n}+1) = (-1,1).$ which is connected

Actually im not able to find the example that $\bigcap\limits _n A_n$ is not connected?

edited Dec 19 '19 at 19:59

Bernard

179,256

asked Dec 19 '19 at 19:57

jasmine

15,021

1

I think you need your sets to be subsets of $\Bbb R^2$. – Lubin Dec 19 '19 at 20:34
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In your attempt you have $A_n\subset\Bbb R$. There is no valid example in $\Bbb R$; the question talks about $A_n\subset \Bbb R^2$. – David C. Ullrich Dec 21 '19 at 16:05
thanks u @DavidC.Ullrich i missed that logics – jasmine Dec 21 '19 at 16:06

find the example that $\bigcap\limits _n A_n$ is not connected?

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