The Stacks project

Lemma 27.29.3. Let $X$ be a quasi-compact scheme. There exists a dense open $V \subset X$ which is separated.

Proof. Say $X = \bigcup _{i = 1, \ldots , n} U_ i$ is a union of $n$ affine open subschemes. We will prove the lemma by induction on $n$. It is trivial for $n = 1$. Let $V' \subset \bigcup _{i = 1, \ldots , n - 1} U_ i$ be a separated dense open subscheme, which exists by induction hypothesis. Consider

\[ V = V' \amalg (U_ n \setminus \overline{V'}). \]

It is clear that $V$ is separated and a dense open subscheme of $X$. $\square$


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