Lemma 5.3.4 (08ZH)—The Stacks project

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Table of contents
Part 1: Preliminaries
Chapter 5: Topology
Section 5.3: Hausdorff spaces
Lemma 5.3.4 (cite)

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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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