Describe closure and interior of subspace $A$ of $X \times B$ in terms of interior and closure of his projection in $X$

Question

Let $X$ any topological space and $B$ a topologic space consisting in $\{0,1\}$ with banal topology. My question is to describe closure and interior of subspace $A$ of $(X \times B)$ in terms of interior and closure of his projection in $X$. At first, I thought to define the projection of $A$ in $X$, but I don't know how define it.

Answer 1

Case 1. A is empty.
Case 2. A = D×{a}, a = 0 or 1.
. . A$^o$ is empty, $\overline A$ = $\overline D$×B.

Case 3. A = D×{0} $\cup$ E×{1}.
. . $\bar A$ = $\overline{D×\{0\}} \cup \overline{E×\{1\}}$
. . = $\bar D×B \cup \bar E×B$
. . = ($\bar D$ $\cup$ $\bar E)×B$
. . = $\overline {D \cup E}×B$.

A$^o$ = (D $\cap$ E)$^o$×B