ag.algebraic geometry - Is the functor of open subschemes representable?

First some simple observations in order to motivate the question:

The functor $Set^{op} to Set, X to {text{subsets of }X}, f to (U to f^{-1}(U))$ is representable. The representing object is ${0,1}$ with the universal subset ${0}$. Also the functor $Top^{op} to Set, X to {text{open subsets of }X}$ is representable: Endow ${0,1}$ with the topology such that ${0}$ is the unique nontrivial open subset (Sierpinski space), then it is again the representing object.

But what about schemes. Is the functor $Sch^{op} to Set, X to {text{open subschemes of }X}$ representable? Of course, we could also talk about open subsets of $X$. My first idea was to endow the Sierpinski space above with a scheme structure, using DVR, but this does not work properly.