= Complete variety
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A <variety> over a field $k$ is complete when its map to $\operatorname{Spec}k$ is <universally closed>: after any base change, projection sends closed subsets to closed subsets. For a separated finite-type <variety>, this is equivalent to being <proper>. In the classical algebraically closed setting one may test projections $X\times Y\to Y$ for arbitrary <varieties>. Images of rational points over a non-algebraically-closed field must not replace geometric images.
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