Let $X$ be a topological space.
For any subsets $A,B \subseteq X,$ $\partial (A \cup B) \subseteq \partial A \cup \partial B.$ In general, they are not equal.
When $A$ and $B$ are disjoint and open, they do seem to be equal. I am not sure how to prove this, though, and I think my intuitive pictures use some sort of separation property in $X$.
Best Answer
Let $A$ and $B$ be open and disjoint. Write $(-)^c$ for the complement of a set. Since $A$ and $B$ are disjoint we have $A\subseteq B^c$ and thus $\partial A\subseteq\bar{A}\subseteq\overline{B^c}=B^c$ since $B^c$ is closed. This means that $\partial A\cap B^c=\partial A$. By replacing $A$ and $B$ we also have $\partial B\cap A^c=\partial B$.
Using this we obtain $$\partial(A\cup B)=\overline{A\cup B}\setminus(A\cup B)=(\bar{A}\cup\bar{B})\cap A^c\cap B^c=((\bar{A}\setminus A)\cap B^c)\cup((\bar{B}\setminus B)\cap A^c)=(\partial A\cap B^c)\cup(\partial B\cap A^c)=\partial A\cup\partial B\,.$$