Lemma 21.31.1. The category $\textit{LC}$ has fibre products and a final object and hence has arbitrary finite limits. Given morphisms $X \to Z$ and $Y \to Z$ in $\textit{LC}$ with $X$ and $Y$ quasi-compact, then $X \times _ Z Y$ is quasi-compact.

Proof. The final object is the singleton space. Given morphisms $X \to Z$ and $Y \to Z$ of $\textit{LC}$ the fibre product $X \times _ Z Y$ is a subspace of $X \times Y$. Hence $X \times _ Z Y$ is Hausdorff as $X \times Y$ is Hausdorff by Topology, Section 5.3.

If $X$ and $Y$ are quasi-compact, then $X \times Y$ is quasi-compact by Topology, Theorem 5.14.4. Since $X \times _ Z Y$ is a closed subset of $X \times Y$ (Topology, Lemma 5.3.4) we find that $X \times _ Z Y$ is quasi-compact by Topology, Lemma 5.12.3.

Finally, returning to the general case, if $x \in X$ and $y \in Y$ we can pick quasi-compact neighbourhoods $x \in E \subset X$ and $y \in F \subset Y$ and we find that $E \times _ Z F$ is a quasi-compact neighbourhood of $(x, y)$ by the result above. Thus $X \times _ Z Y$ is an object of $\textit{LC}$ by Topology, Lemma 5.13.2. $\square$

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