Is any continuous functions between Hausdorff spaces measurable wrt. the Borel algebras?

Question

The title.

Let $A, B$ be Hausdorff. If $f: A\to B$ is continuous, is

$$ f: (A, \mathcal{B}_A)\to (B, \mathcal{B}_B) $$

necessarily measurable?

Answer 1

You don't even need Hausdorff. You just need $A$ and $B$ to be topological spaces. The usual proof that continuous functions from $\mathbb{R}$ to $\mathbb{R}$ are Borel measurable (that it's enough to check measurability on a generating family of your $\sigma$-algebra) just carries over directly.

Answer 2

Yes, this holds for all topological spaces. Let: $$\mathcal{S}=\{ O \subseteq B: f^{-1}[O] \in \mathcal{B}_A \}$$

and check that it is a $\sigma$-algebra in $B$.

By continuity it contains the open subsets of $B$ so $\mathcal{B}_B \subseteq \mathcal{S}$ (as $\mathcal{B}_B$ is the smallest $\sigma$-algebra containing the open sets) and this means that exactly that $$\forall O \in \mathcal{B}_B: f^{-1}[O] \in \mathcal{B}_A$$

or, that $f$ is Borel-measurable.