What are some closed, disjoint subsets $A, B$ in $R^2$ where $inf\{d(A, B) = 0 \forall a \in A \forall b \in B\}$?

  • 5
    $\begingroup$ How about the graph of the equation $xy=1$ and the x-axis? $\endgroup$
    – Old John
    Oct 25 '13 at 20:35
  • 1
    $\begingroup$ Did you try and search the site before posting the question? $\endgroup$
    – Asaf Karagila
    Oct 25 '13 at 20:36
  • $\begingroup$ @OldJohn is it closed by ($0, \infty$)? That's open. Please clarify why that works. $\endgroup$
    – Don Larynx
    Oct 25 '13 at 20:52
  • $\begingroup$ @DonLarynx The x-axis is the set $(-\infty, \infty)$ in $\mathbb{R}^2$, and this is clearly closed. $\endgroup$
    – Old John
    Oct 25 '13 at 20:55
  • $\begingroup$ @OldJohn if it was closed why isn't it denoted by $[\infty, \infty]$? $\endgroup$
    – Don Larynx
    Oct 26 '13 at 14:25

For example, $A=\{(x,y)\mid xy=1,x>0\}$ and $B=\{(x,y)\mid xy=-1,x<0\}$


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