Construction of a set

Question

Let $B,Y$ two closed sets from a Banach space $X$, such that $Y\not\subset [X\setminus \rm int(B)]$, i have to find a set $D\neq\emptyset$ such that :

$D$ is closed,

$D\subset B$ and

$\rm int(D)\subset [X\setminus Y]$

how to find it please ?

Answer 1

The premise is equivalent to $Y \cap \operatorname{int} B$ is not empty.
Let $b$ be any point in $B$. Take $D = \{b\}$.