Lemma 87.19.10. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of affine formal algebraic spaces which is representable by algebraic spaces. Then

if $Y$ is countably indexed, then $X$ is countably indexed,

if $Y$ is countably indexed and classical, then $X$ is countably indexed and classical,

if $Y$ is weakly adic, then $X$ is weakly adic,

if $Y$ is adic*, then $X$ is adic*, and

if $Y$ is Noetherian and $f$ is (locally) of finite type, then $X$ is Noetherian.

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