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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