I have a question about an argument from Mumford's "Red Book of Varieties and Schemes" (page 119). Here the relevant excerpt:
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:
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 $X \times_{Spek(k)} X$.