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Math Help - Two sets intersect => their boundaries also intersect?

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    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!
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    Quote Originally Posted by zzzhhh View Post
    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 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
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