Proof.
Proof of (1). Let x \in \overline{E}. Let \{ U_ i\} be the set of quasi-compact open neighbourhoods of x. A finite intersection of the U_ i is another one. The intersection U_ i \cap E is nonempty for all i. Since the subsets U_ i \cap E are closed in the constructible topology we see that \bigcap (U_ i \cap E) is nonempty by Lemma 5.23.2 and Lemma 5.12.6. Since \{ U_ i\} is a fundamental system of open neighbourhoods of x, we see that \bigcap U_ i is the set of generalizations of x. Thus x is a specialization of a point of E.
Part (2) is immediate from (1).
Proof of (3). Assume E' is as in (3). The complement of E' is closed in the constructible topology (Lemma 5.15.2) and closed under specialization (Lemma 5.19.2). Hence the complement is closed by (2), i.e., E' is open.
\square
Comments (4)
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