# The Stacks Project

## Tag 094L

Lemma 27.2.4. Let $X$ be a quasi-compact and quasi-separated scheme. Then the underlying topological space of $X$ is a spectral space.

Proof. By Topology, Definition 5.23.1 we have to check that $X$ is sober, quasi-compact, has a basis of quasi-compact opens, and the intersection of any two quasi-compact opens is quasi-compact. This follows from Schemes, Lemma 25.11.1 and 25.11.2 and Lemma 27.2.3 above. $\square$

The code snippet corresponding to this tag is a part of the file properties.tex and is located in lines 101–105 (see updates for more information).

\begin{lemma}
\label{lemma-quasi-compact-quasi-separated-spectral}
Let $X$ be a quasi-compact and quasi-separated scheme.
Then the underlying topological space of $X$ is a spectral space.
\end{lemma}

\begin{proof}
By Topology, Definition \ref{topology-definition-spectral-space}
we have to check that $X$ is sober, quasi-compact, has a basis
of quasi-compact opens, and the intersection of any two
quasi-compact opens is quasi-compact. This follows from
Schemes, Lemma \ref{schemes-lemma-scheme-sober} and
\ref{schemes-lemma-basis-affine-opens}
and
Lemma \ref{lemma-quasi-separated-quasi-compact-open-retrocompact}
above.
\end{proof}

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