Lemma 7.17.2. Let $\mathcal{C}$ be a site. Let $U$ be an object of $\mathcal{C}$. Consider the following conditions

1. $U$ is quasi-compact,

2. for every covering $\{ U_ i \to U\} _{i \in I}$ in $\mathcal{C}$ there exists a finite covering $\{ V_ j \to U\} _{j = 1, \ldots , m}$ of $\mathcal{C}$ refining $\mathcal{U}$, and

3. for every covering $\{ U_ i \to U\} _{i \in I}$ in $\mathcal{C}$ there exists a finite subset $I' \subset I$ such that $\{ U_ i \to U\} _{i \in I'}$ is a covering in $\mathcal{C}$.

Then we always have (3) $\Rightarrow$ (2) $\Rightarrow$ (1) but the reverse implications do not hold in general.

Proof. The implications are immediate from the definitions. Let $X = [0, 1] \subset \mathbf{R}$ as a topological space (with the usual $\epsilon$-$\delta$ topology). Let $\mathcal{C}$ be the category of open subspaces of $X$ with inclusions as morphisms and usual open coverings (compare with Example 7.6.4). However, then we change the notion of covering in $\mathcal{C}$ to exclude all finite coverings, except for the coverings of the form $\{ U \to U\}$. It is easy to see that this will be a site as in Definition 7.6.2. In this site the object $X = U = [0, 1]$ is quasi-compact in the sense of Definition 7.17.1 but $U$ does not satisfy (2). We leave it to the reader to make an example where (2) holds but not (3). $\square$

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