Example

Every constant mapping into a Hausdorff space is closed. But it $f$ is the constant mapping $Q \to 0$, then $f \times \iota_Q$ is the mapping $(x, y) \to (0, y)$ of $Q^2$ into $Q^2$, so it is the second projection and is not closed (§ 4, no. 2, Remark 1).