Lemma 5.23.7. Let $X$ be a spectral space. Let $x, y \in X$. Then either there exists a third point specializing to both $x$ and $y$, or there exist disjoint open neighbourhoods containing $x$ and $y$.

Proof. Let $\{ U_ i\}$ be the set of quasi-compact open neighbourhoods of $x$. A finite intersection of the $U_ i$ is another one. Let $\{ V_ j\}$ be the set of quasi-compact open neighbourhoods of $y$. A finite intersection of the $V_ j$ is another one. If $U_ i \cap V_ j$ is empty for some $i, j$ we are done. If not, then the intersection $U_ i \cap V_ j$ is nonempty for all $i$ and $j$. The sets $U_ i \cap V_ j$ are closed in the constructible topology on $X$. By Lemma 5.23.2 we see that $\bigcap (U_ i \cap V_ j)$ is nonempty (Lemma 5.12.6). Since $X$ is a sober space and $\{ U_ i\}$ is a fundamental system of open neighbourhoods of $x$, we see that $\bigcap U_ i$ is the set of generalizations of $x$. Similarly, $\bigcap V_ j$ is the set of generalizations of $y$. Thus any element of $\bigcap (U_ i \cap V_ j)$ specializes to both $x$ and $y$. $\square$

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