I don't understand how sets can be closed, yet disjoint? [duplicate]

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

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

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