Borel measurable function

For topological spaces $X,Y$, a Borel measurable function $f:X\to Y$ is a measurable function for the Borel sigma-algebras on both spaces. Equivalently, the preimage of every open set in $Y$ is a Borel set in $X$. With a different source sigma-algebra $\mathcal{A}$, the phrase measurable into $Y$ with its Borel sigma-algebra means $f^{-1}(B)\in \mathcal{A}$ for every Borel $B\subseteq Y$; specifying the source is essential.