TITLE: Is it true that if two open sets in a topological space intersect each other and share the same boundary, they are the same?
QUESTION [9 upvotes]: Suppose we have a topological space $(X,\mathcal T)$. Open sets $A,B$ have the following properties:
$A\cap B\ne\emptyset$
$\partial A=\partial B$
Then $A=B$. Is it correct? If so, how to prove it?

REPLY [2 votes]: Let $A=B \cup C$ where $C$ is open and closed, $B$ is open, $B$ and $C$ are disjoint and neither $B$ nor $C$ are empty. Then $A$ and $B$ have the same boundary.