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 f : dom ( f ) → R ¯ taking values in the projective line R ¯ = R ∪ { ∞ } and defined on a cofinite subset dom ( f ) of the torus R ¯ 2 . Then the family F determines a continuous vector-function F : dom ( F ) → R ¯ F defined on the dense G δ -set dom ( F ) = ⋂ f ∈ F dom ( F ) of R ¯ 2 . The closure Γ ¯ ( F ) of its graph Γ ( F ) = { ( x , f ( x ) ) : x ∈ dom ( F ) } in R ¯ 2 × R ¯ F is called the graphoid of the family F . We prove that the graphoid Γ ¯ ( F ) has topological dimension dim ( Γ ¯ ( F ) ) = 2 . If the family F contains all linear fractional transformations f ( x , y ) = x − a y − b for ( a , b ) ∈ Q 2 , then the graphoid Γ ¯ ( F ) has cohomological dimension dim G ( Γ ¯ ( F ) ) = 1 for any non-trivial 2-divisible abelian group G. Hence the space Γ ¯ ( F ) is a natural example of a compact space that is not dimensionally full-valued and by this property resembles the famous Pontryagin surface.