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Let $X$ be a scheme, let $D$ be a very dense subset of $X$ and let $Y$ be a pro-constructible subset of $X$. Is it true that $Y \cap D \neq \emptyset$?

If $Y$ is just constructible, this is true.

What I would like to see is that the map $Y \mapsto Y \cap D$ yields a bijection between pro-constructible (ind-constructible) subsets of $X$ and $D$. Again, this is true for constructibles.

(You can assume that $X$ is Noetherian, Jacobson, or even more; of finite type over a field is fine, too. Also you can assume that $D$ is the set of closed points in this case.)

A somewhat related question is: Does an ind-constructible subset of $X$ containing all closed points of $X$ also contain the generic point of $X$? Is it already equal to $X$?

Thanks.

Edit: Perhaps there are very stupid counterexamples to my question. I have a bad intuition here. What about the more restrictive case that $Y$ is the complement of a (possibly infinite) union of locally closed sets.

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  • $\begingroup$ If $X$ is integral and Jacobson, the generic point is pro-constructible and the set of closed points is very dense. $\endgroup$
    – Laurent Moret-Bailly
    Commented Feb 26, 2015 at 17:27
  • $\begingroup$ Yes, correct, thanks. This is somehow the stupid counter-example that came to my mind only after asking the question. What about the restriction mentioned in the edit? Is the complement of the generic point a union of locally closed sets? $\endgroup$
    – user68570
    Commented Feb 26, 2015 at 19:23
  • $\begingroup$ @user68750: It is the union of all proper closed subsets. $\endgroup$
    – Laurent Moret-Bailly
    Commented Feb 26, 2015 at 22:29
  • $\begingroup$ Yes, alright, so it's not true. This stuff is not preserved under quasi-homemorphisms... $\endgroup$
    – user68570
    Commented Feb 27, 2015 at 10:18

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