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?
Edit: A side question. If we have a map $f:X\to Y$ of -say- varieties, we can put a closed subscheme structure on the (Zariski) closure $Z$ of $f(X)$, as described in Hartshorne's book. On the other hand we can consider the ringed space (that will not be, in genberal, a scheme) $W$ which is the quotient of $X$ by the equivalence relation induced by $f$. Will there be any relation between $Z$ and $W$?