# The Stacks Project

## Tag 07ZL

Lemma 27.2.6. Let $X$ be a scheme. A subset $Z$ of $X$ is retrocompact in $X$ if and only if $E \cap U$ is quasi-compact for every affine open $U$ of $X$.

Proof. Immediate from the fact that every quasi-compact open of $X$ is a finite union of affine opens. $\square$

The code snippet corresponding to this tag is a part of the file properties.tex and is located in lines 137–141 (see updates for more information).

\begin{lemma}
\label{lemma-retrocompact}
Let $X$ be a scheme. A subset $Z$ of $X$ is retrocompact in $X$ if and only if
$E \cap U$ is quasi-compact for every affine open $U$ of $X$.
\end{lemma}

\begin{proof}
Immediate from the fact that every quasi-compact open of $X$ is a finite
union of affine opens.
\end{proof}

Comment #3199 by Dennis Keeler on February 11, 2018 a 6:14 pm UTC

I believe that $Z=E$ is intended here.

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