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Plesae give some hint to solve the following problem:

If $A$ and $B$ are two subsets of a topological space $X$ such that $\overline{A}\cap \overline{B}=\emptyset$, then $\partial(A\cup B)=\partial A\cup \partial B$, where $\partial A$ denotes the boundary of $A$.

Anupam
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  • I think it would suffice to prove that $\overline(A\cup B) = \overline A \cup \overline B$ and $(A\cup B)^\circ = A^\circ \cup B^\circ$ (where ${}^\circ$ denotes the interior). Can you derive either of these from the given assumption? (Intuitively, $\overline A \cap \overline B = \emptyset$ means that the sets $A$ and $B$ stay well separated from each other. It implies, for example, that every point of $A$ has a neighborhood disjoint from $B$ and vice versa.) – Greg Martin Oct 02 '13 at 06:00
  • I got stuck exactly at this point...how to show that $\overline{A\cup B}=\overline{A}\cup \overline{B}$? – Anupam Oct 02 '13 at 06:19
  • This is a good example of why you should include your progress when you post questions. If you had included where you got stuck, you probably would have gotten a helpful response much sooner. – Greg Martin Oct 02 '13 at 18:53

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