In the past by the term "variety" people understood a subset of projective space locally closed for the Zariski topology. Now we have a more natural notion of abstract algebraic variety, i.e. a scheme that is so and so, and we can conceive non quasi-projective varieties.

Now we have the concept of constructible subset of a variety (or of a scheme), i.e. a finite union of locally closed subsets (subschemes). We know that the image of a morphism of varieties may fail to be a subvariety of the target, nevertheless it's always a constructible subset thereof.

Is there a reasonable notion of "abstract constructible space"? And would it be of any utility in algebraic geometry?