Let $f : X \to X/\sim$ be a quotient map from a topological space $X$ to the quotient space $X/\sim$ for $\sim$ some equivalence relation. Let $S \subseteq X/\sim$. Is it true that $f^{-1}(\overline{S}) = \overline{f^{-1}(S)}$?

The specific case I have in mind is a Borel subgroup $B$ of a Chevalley group $G$ acting on $G$. The Bruhat decomposition decomposes $G$ into the disjoint union of Bruhat cells $BwB$ over representatives $w$ of the elements of the Weyl group $N(T)/T$ where $T$ is a maximal torus in $B$ in $N(T)$ is it's normalizer in $G$. Taking the quotient under the action of $B$ on $G$ induces a decomposition of $G/B$ into the disjoint union of $Bw.B$ where $Bw.B$ is the set of cosets of the form $bwB$. The closures of the Bruhat cells $Bw.B$ in $G/B$ are unions of Bruhat cells and I want to know if this therefore implies that the closures of the corresponding Bruhat cells in $G$ are also unions of Bruhat cells.