Two sets intersect => their boundaries also intersect?

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March 1st 2010, 02:20 PM
zzzhhh

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!
March 1st 2010, 02:23 PM
Drexel28

Quote:

Originally Posted by zzzhhh

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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