Dimension of graphoids of rational vector-functions

Abstract

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 $G_\delta$-set $dom(F)=\bigcap_{f\in F}dom(F)$ of $\bar R^2$. The closure $\bar\Gamma(F)$ of its graph $\Gamma(F)=\{(x,f(x)):x\in dom(F)\}$ in $\bar R^2\times\bar R^F$ is called the {\em graphoid} of the family $F$. We prove the graphoid $\bar\Gamma(F)$ has topological dimension $dim(\bar\Gamma(F))=2$. 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 $\bar\Gamma(F)$ has cohomological dimension $dim_G(\bar\Gamma(F))=1$ for any non-trivial 2-divisible abelian group $G$. Hence the space $\bar\Gamma(F)$ is a natural example of a compact space that is not dimensionally full-valued and by this property resembles the famous Pontryagin surface.Comment: 20 page