Two sets intersect => their boundaries also intersect?
In a topological space, if two subset A and B intersect, do their boundaries also intersect? If not, under what conditions can this statement be true? Thanks!
In a topological space, if two subset A and B intersect, do their boundaries also intersect? If not, under what conditions can this statement be true? Thanks!
What about $\displaystyle A=(0,1),B=\left[\tfrac{1}{2},\tfrac{3}{4}\right]$. We see that $\displaystyle A\cap B\ne \varnothing$ but $\displaystyle \partial A\cap\partial B=\{0,1\}\cap\left\{\tfrac{1}{2},\tfrac{3}{4}\righ t\}=\varnothing$