Woops! I'll add explicitly that I want algebraic functions.

@Mariano: a reasonable definition of regular commutative rings exists in the non-Noetherian setting: every finitely presented module admits a finite projective resolution (but you don't require projective dimension to be bounded, of course). Then a filtered colimit of regular rings under flat maps is again regular.

I think that $|X|$ is fairly standard notation for the set of closed points of a scheme $X$. E.g., Deligne uses it in Weil I and II.

$|\operatorname{Spec}(\mathbb{Z})|$.