Lemma 5.3.4. Let $X \to Z$ and $Y \to Z$ be continuous maps of topological spaces. If $Z$ is Hausdorff, then $X \times _ Z Y$ is closed in $X \times Y$.

**Proof.**
This follows from Lemma 5.3.1 as $X \times _ Z Y$ is the inverse image of $\Delta (Z)$ under $X \times Y \to Z \times Z$.
$\square$

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