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  "body": "\n## Definition\n\nLet $f \\colon X \\to Y$ be a [[morphism of schemes|JXRKH6P1Z4]].\nThe morphism $f$ is *universally closed* if the following holds for every morphism of schemes $Y' \\to Y$.\nThe base change\n$$f' \\colon X \\times_Y Y' \\to Y'$$\nis a [[closed map|00FSJTXPJM]] of the underlying topological spaces.\nHere $X \\times_Y Y'$ is the [[fiber product|1DZMT4S9FB]] and $f'$ is the second projection.\n\n## Remarks\n\nA closed immersion $Z \\to Y$ is universally closed.\nIts base change along any $Y' \\to Y$ is again a closed immersion, and a closed immersion is a homeomorphism onto a closed subset, so it is a closed map.\nFor a field $k$, the morphism $\\mathbb{A}^1_k \\to \\operatorname{Spec} k$ is closed but not universally closed.\nIts base change along $\\mathbb{A}^1_k \\to \\operatorname{Spec} k$ is the projection $\\mathbb{A}^2_k \\to \\mathbb{A}^1_k$, which sends the closed set $V(xy - 1)$ onto $\\mathbb{A}^1_k \\setminus \\set{0}$, a set that is not closed.\nUniversally closed morphisms are stable under composition and base change.\n",
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