Definition (Borel Measurable Function)

Suppose $(X,\mathscr{T}_{1})$ and $(Y,\mathscr{T}_{2})$ are two Topological Spaces, and $f:X\to Y$ is a function. Let $\mathcal{B}$ be the Borel σ-algebra associated with the Topology $\mathscr{T}_{1}$. If we have $$\forall B\in \mathcal{B}(Y):f^{-1}(B)\in \mathcal{B}(X)$$or $$\forall \mathcal{O}\in\mathscr{T}_{2}:\quad f^{-1}(\mathcal{O})\in\mathcal{B}(\mathscr{T}_{1})$$ then we say that $f$ is a Borel-Measurable Function or simply a Borel function.