Lemma 85.4.12. Let $\varphi : A \to B$ be a continuous map of weakly admissible topological rings. The following are equivalent

1. $\varphi$ is taut,

2. for every weak ideal of definition $I \subset A$ the closure of $\varphi (I)B$ is a weak ideal of definition of $B$ and these form a fundamental system of weak ideals of definition of $B$.

Proof. It is clear that (2) implies (1). The other implication follows from Lemma 85.4.10. $\square$

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