$X=\overset{\circ}{(X\setminus U)}\cup \overset{\circ}{B}$ if $\bar U\subseteq \overset{\circ}{B}$? Let $X$ be a topological space and $U,B\subset X$ two subspaces of $X$ with the property that the closure of $U$ is c...--prophetes.ai

$X=\overset{\circ}{(X\setminus U)}\cup \overset{\circ}{B}$ if $\bar U\subseteq \overset{\circ}{B}$? Let $X$ be a topological space and $U,B\subset X$ two subspaces of $X$ with the property that the closure of $U$ is contained in the interior of $B$, i.e. $\bar U\subseteq \overset{\circ}{B}$. > Is it true that $X=\overset{\circ}{(X\setminus U)}\cup \overset{\circ}{B}$ ?

Yes, it is. The interior of the complement is the complement of the closure.