The Stacks project

Lemma 21.31.3. Let $X$ be a Hausdorff and locally quasi-compact space, in other words, an object of $\textit{LC}$.

  1. If $X' \to X$ is an isomorphism in $\textit{LC}$ then $\{ X' \to X\} $ is a qc covering.

  2. If $\{ f_ i : X_ i \to X\} _{i\in I}$ is a qc covering and for each $i$ we have a qc covering $\{ g_{ij} : X_{ij} \to X_ i\} _{j\in J_ i}$, then $\{ X_{ij} \to X\} _{i \in I, j\in J_ i}$ is a qc covering.

  3. If $\{ X_ i \to X\} _{i\in I}$ is a qc covering and $X' \to X$ is a morphism of $\textit{LC}$ then $\{ X' \times _ X X_ i \to X'\} _{i\in I}$ is a qc covering.

Proof. Part (1) holds by the remark above that open coverings are qc coverings.

Proof of (2). Let $x \in X$. Choose $i_1, \ldots , i_ n \in I$ and $E_ a \subset X_{i_ a}$ quasi-compact such that $\bigcup f_{i_ a}(E_ a)$ is a neighbourhood of $x$. For every $e \in E_ a$ we can find a finite subset $J_ e \subset J_{i_ a}$ and quasi-compact $F_{e, j} \subset X_{ij}$, $j \in J_ e$ such that $\bigcup g_{ij}(F_{e, j})$ is a neighbourhood of $e$. Since $E_ a$ is quasi-compact we find a finite collection $e_1, \ldots , e_{m_ a}$ such that

\[ E_ a \subset \bigcup \nolimits _{k = 1, \ldots , m_ a} \bigcup \nolimits _{j \in J_{e_ k}} g_{ij}(F_{e_ k, j}) \]

Then we find that

\[ \bigcup \nolimits _{a = 1, \ldots , n} \bigcup \nolimits _{k = 1, \ldots , m_ a} \bigcup \nolimits _{j \in J_{e_ k}} f_ i(g_{ij}(F_{e_ k, j})) \]

is a neighbourhood of $x$.

Proof of (3). Let $x' \in X'$ be a point. Let $x \in X$ be its image. Choose $i_1, \ldots , i_ n \in I$ and quasi-compact subsets $E_ j \subset X_{i_ j}$ such that $\bigcup f_{i_ j}(E_ j)$ is a neighbourhood of $x$. Choose a quasi-compact neighbourhood $F \subset X'$ of $x'$ which maps into the quasi-compact neighbourhood $\bigcup f_{i_ j}(E_ j)$ of $x$. Then $F \times _ X E_ j \subset X' \times _ X X_{i_ j}$ is a quasi-compact subset and $F$ is the image of the map $\coprod F \times _ X E_ j \to F$. Hence the base change is a qc covering and the proof is finished. $\square$

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