 Solution

If $x\in A=g^{-1}(B)$ and $y\mathcal{R}x$, the definition of the relation applies in particular to the fixed continuous map $g$ into the fixed Hausdorff space $Y$. It gives $g(y)=g(x)\in B$, hence $y\in A$. Thus $A$ is a union of whole equivalence classes, also called a saturation of a subset.

Always $A\subseteq p^{-1}(p(A))$. Conversely, a point in $p^{-1}(p(A))$ shares its class with some $x\in A$, and the preceding argument places it in $A$. Hence