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How can I find an example of subsets which satisfies;

for every $i \in \mathbb{N}$
$c_1 \supset c_2 \supset c_3 \supset \cdots \supset c_n$ which are closed and connected subsets of $\mathbb{R^k}$ then $\bigcap_{i=1}^\infty c_i$ is not connected.

until now I have

  • $c_n=\{[n,\infty) , n \in \mathbb{N}\}$ $\Rightarrow$ $\bigcap_{i=1}^\infty c_i=\varnothing$ which is I couldn't decide whether if it is connected or not connected.
  • $d_n=\{ ( [-\frac{1}{n},\frac{1}{n}] \times \{0\}) \cup (\{0\}\times ([-1,1]\setminus \{0\})), n\in \mathbb{N}\}$ $\Rightarrow$ $\bigcap_{i=1}^\infty d_i= \{0\} \times [-1,1]$ which is connected.
    (here I want to seperate the vertical one from the point $(0,0)$ for not path connected implies not connected. If i rearrange the $ [-1,1] \times \{0\}$ element -for not path connecting result- as $(-1,1) \times \{0\}$ the set will be not closed but intersection is not connected.)

Thanks in advance for your guidances :)

Lloth
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    Empty sets are connected (vacuously), so this causes problems with $c_n$. With $d_n$, where is the dependence on $n$? – Michael Burr May 11 '17 at 11:07

2 Answers2

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So, you've got some good ideas here. Combine them and you're done. It's hard to give a hint, so here's an example:

$$e_n=(\mathbb{R}\times\{0\})\cup (\mathbb{R}\times\{1\})\cup\{(x,y):x\geq n,0\leq y\leq 1\}.$$

Now, this consists of two horizontal lines and part of the region between them. It's kinda like your first example because the intersection of the region between the lines is empty.

Michael Burr
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  • Thank you so much, The example that you gave is really beneficial for me cause it is combination of my non-true examples. i am greatful to you :) – Lloth May 11 '17 at 11:19
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Another example. Let $c_n =\mathbb{R}^2 \setminus \{(x,y)| x \in (-1,1),y \in (-n,n)\}$. Complement is open so $c_n$ is closed. They are clearly nested and $c = \cap_i^\infty c_n = \mathbb{R}^2\setminus((-1,1)\times Y) $ is disconnected.

ElChorro
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