Lemma 87.10.4. Let $S$ be a scheme. Let $X$ be a presheaf on $(\mathit{Sch}/S)_{fppf}$. The following are equivalent

$X$ is a countably indexed affine formal algebraic space,

$X = \text{Spf}(A)$ where $A$ is a weakly admissible topological $S$-algebra which has a countable fundamental system of neighbourhoods of $0$,

$X = \text{Spf}(A)$ where $A$ is a weakly admissible topological $S$-algebra which has a fundamental system $A \supset I_1 \supset I_2 \supset I_3 \supset \ldots $ of weak ideals of definition,

$X = \text{Spf}(A)$ where $A$ is a complete topological $S$-algebra with a fundamental system of open neighbourhoods of $0$ given by a countable sequence $A \supset I_1 \supset I_2 \supset I_3 \supset \ldots $ of ideals such that $I_ n/I_{n + 1}$ is locally nilpotent, and

$X = \text{Spf}(A)$ where $A = \mathop{\mathrm{lim}}\nolimits B/J_ n$ with the limit topology where $B \supset J_1 \supset J_2 \supset J_3 \supset \ldots $ is a sequence of ideals in an $S$-algebra $B$ with $J_ n/J_{n + 1}$ locally nilpotent.

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