Lemma 5.16.4. Let $X$ be a Noetherian topological space. Let $x \in X$. Let $E \subset X$ be constructible in $X$. The following are equivalent:

1. $E$ is a neighbourhood of $x$, and

2. for every irreducible closed subset $Y$ of $X$ which contains $x$ the intersection $E \cap Y$ is dense in $Y$.

Proof. It is clear that (1) implies (2). Assume (2). Consider the set $\mathcal{S}$ of closed subsets $Y$ of $X$ containing $x$ such that $E \cap Y$ is not a neighbourhood of $x$ in $Y$. If $\mathcal{S} \not= \emptyset$, then it has a minimal element $Y$ as $X$ is Noetherian. Suppose $Y = Y_1 \cup Y_2$ with two smaller nonempty closed subsets $Y_1$, $Y_2$. If $x \in Y_ i$ for $i = 1, 2$, then $Y_ i \cap E$ is a neighbourhood of $x$ in $Y_ i$ and we conclude $Y \cap E$ is a neighbourhood of $x$ in $Y$ which is a contradiction. If $x \in Y_1$ but $x \not\in Y_2$ (say), then $Y_1 \cap E$ is a neighbourhood of $x$ in $Y_1$ and hence also in $Y$, which is a contradiction as well. We conclude that $Y$ is irreducible closed. By assumption (2) we see that $E \cap Y$ is dense in $Y$. Thus $E \cap Y$ contains an open $V$ of $Y$, see Lemma 5.16.3. If $x \in V$ then $E \cap Y$ is a neighbourhood of $x$ in $Y$ which is a contradiction. If $x \not\in V$, then $Y' = Y \setminus V$ is a proper closed subset of $Y$ containing $x$. By minimality of $Y$ we see that $E \cap Y'$ contains an open neighbourhood $V' \subset Y'$ of $x$ in $Y'$. But then $V' \cup V$ is an open neighbourhood of $x$ in $Y$ contained in $E$, a contradiction. This contradiction finishes the proof of the lemma. $\square$

Comment #672 by Wei Xu on

There is a small gap in the proof, from line 2293 to line 2297: "Let $Y = Y_1 \cup \ldots \cup Y_r$ be the decomposition of $Y$ into its irreducible components, see Lemma 5.9.2. If $r > 1$, then each $Y_i \cap E$ is a neighbourhood of $x$ in $Y_i$ by minimality of $Y$. "

We can't conclude that EACH
$Y_i \cap E$ is a neighbourhood of $x$ in $Y_i$ since we don't know if $x\in Y_i$ for every $i$.

One possible revision could be: "Suppose $Y=Y_1\cup Y_2$ with two smaller (nonempty) closed subsets $Y_1,Y_2$". If $x\in Y_i$ for $i=1,2$, then $Y_i\cap E$ is a neighbourhood of $x$ in $Y_i$, one can conclude $Y\cap E$ is a neighbourhood of $x$ in $Y$. If $x\in Y_1$ but $x\notin Y_2$ (say), then there is an open $U_1$ of $Y$ such that $x\in U_1\cap Y_1\subset Y_1\cap E$, then $x\in U_1\cap (Y\setminus Y_2)\subset Y\cap E$, we also get that $Y\cap E$ is a neighbourhood of $x$ in $Y$. This is a contradiction, so $Y$ is irreducible closed.....

Besides, I don't know whether the word "smallest element" replaced by "minimal element" would be better?

Thanks.

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