The Stacks project

Proof. Since $X$ and $Y$ are affine it is clear that conditions (1) and (2) are equivalent. In cases (1) and (2) the morphism $f$ is representable by algebraic spaces by definition, hence affine by Lemma 87.19.7. Thus if (1) or (2) holds we see that $X$ is countably indexed by Lemma 87.19.9. Write $X = \text{Spf}(A)$ and $Y = \text{Spf}(B)$ for topological $S$-algebras $A$ and $B$ in $\textit{WAdm}^{count}$, see Lemma 87.10.4. By Lemma 87.9.10 we see that $f$ corresponds to a continuous map $B \to A$. Hence now the result follows from Lemma 87.29.2. $\square$