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 $di{m}_G(\bar{\Gamma }(F))=1$ 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.

20 pages

References in corpus (1)

• Cohomological dimension theory of compact metric spaces