TITLE: If sets $A, B$ in Euclidean space are closed sets, they have the same boundary and their interior's intersection is non-empty, can we say $A=B$?
QUESTION [6 upvotes]: If sets $A, B$ in Euclidean space are closed sets, they have the same boundary and their interiors intersection is non-empty, can we say $A=B$? Any suggestions and comments are welcome!

REPLY [3 votes]: Counterexample in $\mathbb R^2$:
                                               
\begin{align*}
A&=\text{red disk}\bigcup\text{black vertical line}\\
B&=\text{red disk}\bigcup\text{black vertical line}\bigcup\text{blue area}
\end{align*}
The red disk includes the circle on its boundary, and the blue area extends to infinity towards north, south, and east.
\begin{align*}
\partial A&=\partial B=\text{red circle}\bigcup\text{black vertical line}\\
A^{\circ}&=\text{interior of red disk}\\
B^{\circ}&=\text{interior of red disk}\bigcup\text{blue area}\\
A^{\circ}\cap B^{\circ}&=\text{interior of red disk}\neq\varnothing\\
A&\neq B
\end{align*}