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I have a question about an argument from Mumford's "Red Book of Varieties and Schemes" (page 119). Here the relevant excerpt:

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Here we denote by $Z \subset X \times_{Spek(k)}\ X$ the set consists of "points" $z$ with $f(x)\equiv g(x)$. The "$\equiv$"-relation is defined at page 118 as follows:

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From Prop. 4 we know that $Z$ is a locally closed subset of $X \times_{Spek(k)}\ X$. Denote by $Cl := \{x \in X \times_{Spek(k)} X \vert x \text{ is a closed point in } X \times_{Spek(k)} X \}$ the set of closed points.

My QUESTION is that if we assume that the intersection $Z \cap Cl$ is closed in $Cl$ why then this already implies that $Z$ is closed in $X \times_{Spek(k)} X$ as stated in the excerpt?

My considerations: I know that $Cl$ is dense in $X \times_{Spek(k)} X$.

user267839
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  • I do not completely understand the down-votes. The question seems to be fairly specific. One point of possible misunderstanding: you say "the intersection $Z \cap Cl$ is closed in $Cl$). I am not sure Mumford is saying this (he says "in induced topology" but does not specify the superset). I am not sure whether this matters or not. –  Apr 30 '19 at 17:48
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    Possible duplicate of A condition for a locally closed subset of a $k$-scheme of finite type to be closed – KReiser Apr 30 '19 at 18:04
  • @schematic_boi: yes this is a good point: I assumed that by "induced topology" Mumford means the subspace topology of $Cl \subset X \times_{Spek(k)}\ X$. If my conjecture is wrong then this of course matters. What do you think he means here by "induced topology"? – user267839 May 01 '19 at 16:36
  • @KarlPeter I mean, there is an induced topology on any subset of a topological space. Given that here we have two subsets it might be either $Z$ or $Cl$. –  May 01 '19 at 16:37
  • @schematic_boi: so you wanted to point out the ambiguity? hm, since Mumford didn't explicitely mentioned it that seems that it would be obvious (for him) which one meaned. I supposed that refers to $Cl$ since seems to me that it looks like a general topological statement: $A \subset X$ is closed in $X$ iff $A$ is locally closed and closed in a dense set $U $ of $X$. Here $U=Cl$. But I'm not sure. Do you have an idea? – user267839 May 01 '19 at 16:46

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