The distance between two sets $S_1,S_2\subset R^n$ is :
$$d(S_1,S_2) := \underset{x_1\in S_1,x_2\in S_2}{inf} d(x_1,x_2)$$
Can you give me some examples of two closed sets$S_1,S_2\subset R^n$ having no points in common for which $d(S_1,S_2)=0$? I have difficulties in describing such sets.Appreciated!
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Did you search for a duplicate? A standard example: $\mathbb N$ and ${n+\frac 1n, n >1}$. – Kavi Rama Murthy Apr 15 '24 at 04:53
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@geetha290krm Thank you for the reminder,I will pay attention in the future.(sry) – Toboraton Apr 15 '24 at 05:16
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$S_1=\{(x,y):y\le0\}$ and $S_2=\{(x,y):y\ge e^x\}$ the distance is 0 because they "meet at infinity".
Liding Yao
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Let $e_1=(1,0,0,\dots,0)$ and
$$ S_1 = \left\{ne_1\ : n\in\mathbb N \right\} \\ S_2 = \left\{\left(n+\frac{1}{2n}\right)e_1\ : n\in\mathbb N \right\} $$
Basically just take two sequences in $\mathbb R$ that both go to infinity and also get close to each other. To get an example in $\mathbb R^n$ put them on a one dimensional subspace.
ploosu2
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