# Two sets intersect => their boundaries also intersect?

What about $A=(0,1),B=\left[\tfrac{1}{2},\tfrac{3}{4}\right]$. We see that $A\cap B\ne \varnothing$ but $\partial A\cap\partial B=\{0,1\}\cap\left\{\tfrac{1}{2},\tfrac{3}{4}\righ t\}=\varnothing$