Proposition 10

Let $f$ be a continuous mapping of a topological space $X$ into a topological space $Y$. Then the inverse image under $f$ of every Borel set in $Y$ is a Borel set in $X$.

Let $\mathfrak{I}$ be the set of all subsets $A$ of $Y$ such that $f^{-1}(A)$ is a Borel set in $X$. It follows immediately that $\mathfrak{I}$ is a $\sigma$-algebra which contains all the closed subsets of $Y$; hence $\mathfrak{I}$ contains all Borel sets in $Y$.