The Stacks project

Lemma 87.10.1. Let $S$ be a scheme. Let $X$ be an affine formal algebraic space over $S$. The following are equivalent

  1. there exists a system $X_1 \to X_2 \to X_3 \to \ldots $ of thickenings of affine schemes over $S$ such that $X = \mathop{\mathrm{colim}}\nolimits X_ n$,

  2. there exists a choice $X = \mathop{\mathrm{colim}}\nolimits X_\lambda $ as in Definition 87.9.1 such that $\Lambda $ is countable.

Proof. This follows from the observation that a countable directed set has a cofinal subset isomorphic to $(\mathbf{N}, \geq )$. See proof of Algebra, Lemma 10.86.3. $\square$

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