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Let $\{Z_i\}$ a directed projective system of quasi-compact topological spaces with projective limit $Z$. Assume we are given open subsets $U_i \subseteq Z_i$ such that:

1) For every $i \leq j$, the preimage $(Z_j \to Z_i)^{-1}(U_i)$ is contained in $U_j$.

2) For every $i$, the preimage $(Z \to Z_i)^{-1}(U_i)$ equals $Z$.

Does it follow that $U_i = Z_i$ for some $i$?

If not, assume that $\{Z_i\}$ is actually a system of affine schemes and scheme morphisms. Then it is true by (EGA IV, Corollaire 8.3.4). But I wonder if there is a direct proof which avoids all these nasty lemmas about ind-constructible sets ...

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