Dimension of graphoids of rational vector-functions

Let F⊂R(x,y) be a countable family of rational functions of two variables with real coefficients. Each rational function f∈F can be thought as a continuous function taking values in the projective line and defined on a cofinite subset dom(f) of the torus . Then the family F determines a continuous vector-function defined on the dense Gδ-set dom(F)=⋂f∈Fdom(F) of . The closure of its graph Γ(F)={(x,f(x)):x∈dom(F)} in is called the graphoid of the family F. We prove that the graphoid has topological dimension . If the family F contains all linear fractional transformations for (a,b)∈Q2, then the graphoid has cohomological dimension for any non-trivial 2-divisible abelian group G. Hence the space is a natural example of a compact space that is not dimensionally full-valued and by this property resembles the famous Pontryagin surface.