Dimension of graphoids of rational vector-functions

Let $F$ be a countable family of rational functions of two variables with real coefficients. Each rational function $f\in F$ can be thought as a continuous function $f:dom(f)\to\bar R$ taking values in the projective line $\bar R=R\cup\{\infty\}$ and defined on a cofinite subset $dom(f)$ of the torus $\bar R^2$. Then the family $\F$ determines a continuous vector-function $F:dom(F)\to\bar R^F$ defined on the dense $G_δ$-set $dom(F)=\bigcap_{f\in F}dom(F)$ of $\bar R^2$. The closure $\barΓ(F)$ of its graph $Γ(F)=\{(x,f(x)):x\in dom(F)\}$ in $\bar R^2\times\bar R^F$ is called the {\em graphoid} of the family $F$. We prove the graphoid $\barΓ(F)$ has topological dimension $dim(\barΓ(F))=2$. If the family $F$ contains all linear fractional transformations $f(x,y)=\frac{x-a}{y-b}$ for $(a,b)\in Q^2$, then the graphoid $\barΓ(F)$ has cohomological dimension $dim_G(\barΓ(F))=1$ for any non-trivial 2-divisible abelian group $G$. Hence the space $\barΓ(F)$ is a natural example of a compact space that is not dimensionally full-valued and by this property resembles the famous Pontryagin surface.

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