Two closed subsets $A$ and $B$ in $\mathbb{R}$ with $d(A,B)=0$

Question

I am looking for two closed subsets A and B (with $A\cap B = \emptyset$) of $\mathbb{R}$ with $d(A,B)=0$. I found a solution in $\mathbb{R}^2$, namely $A=\{(x,\frac{1}{x})\mid x>0\}$ and $B=\{(x,0)\mid x>0\}$. I know that those subsets have to be unbounded because there is a theorem that says: the distance between two closed subsets of which one is bounded, is greater than 0.

score 3 · Accepted Answer · answered May 28 '13 at 16:18

3

Set $A=\mathbb{N}\setminus \{1\}$ and $B=\{x+\frac{1}{x}:x\in A\}$

answered May 28 '13 at 16:18

vadim123

82,796

Isn't ${x:x\in T}$ simply $T$? – celtschk May 28 '13 at 16:21
@celtschk, okay, if you like. – vadim123 May 28 '13 at 16:22

score 1 · Answer 2 · answered May 28 '13 at 18:47

1

I used one of your sets to give another example: $$A=\{(x,y)\mid xy\ge -1,x<0\},~~B=\{(x,y)\mid xy\ge1,x>0\}$$

enter image description here

answered May 28 '13 at 18:47

Mikasa

67,374

Very nice graphs you draw. You must have a handy scanner? Or camera? – amWhy May 29 '13 at 00:29
@amWhy: It is really a camera! 12 mega pix Canon – Mikasa May 29 '13 at 04:18
awesome! It does good work for you! – amWhy May 29 '13 at 04:19

Two closed subsets $A$ and $B$ in $\mathbb{R}$ with $d(A,B)=0$

2 Answers2