The Stacks project

Lemma 5.14.5. Let $\mathcal{I}$ be a category and let $i \mapsto X_ i$ be a diagram over $\mathcal{I}$ in the category of topological spaces. If each $X_ i$ is quasi-compact and Hausdorff, then $\mathop{\mathrm{lim}}\nolimits X_ i$ is quasi-compact.

Proof. Recall that $\mathop{\mathrm{lim}}\nolimits X_ i$ is a subspace of $\prod X_ i$. By Theorem 5.14.4 this product is quasi-compact. Hence it suffices to show that $\mathop{\mathrm{lim}}\nolimits X_ i$ is a closed subspace of $\prod X_ i$ (Lemma 5.12.3). If $\varphi : j \to k$ is a morphism of $\mathcal{I}$, then let $\Gamma _\varphi \subset X_ j \times X_ k$ denote the graph of the corresponding continuous map $X_ j \to X_ k$. By Lemma 5.3.2 this graph is closed. It is clear that $\mathop{\mathrm{lim}}\nolimits X_ i$ is the intersection of the closed subsets

\[ \Gamma _\varphi \times \prod \nolimits _{l \not= j, k} X_ l \subset \prod X_ i \]

Thus the result follows. $\square$

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