universally closed

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universally closed

  • A morphism \(f: X\to Y\) is universally closed iff the following is a closed map for all \(Z\): \begin{align*} X\times Z\xrightarrow{(f, \operatorname{id}_Z)} Y\times Z \end{align*}
    • Equivalently when \(Y\) is Hausdorff: for any map \(Z\to Y\), the following pullback is a closed map: \begin{align*} X \underset{\scriptscriptstyle {Y} }{\times} Z\to Z \end{align*}
  • For \(X\) Hausdorff and \(Y\) locally compact, \(f:X\to Y\) is universally closed iff \(f\) is proper.

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