VSE knjižnice (vzajemna bibliografsko-kataložna baza podatkov COBIB.SI)
  • Dimension of graphoids of rational vector-functions
    Potyatynyk, Oles ; Banakh, Taras, 1968-
    Let ▫${\mathcal F}$▫ be a countable family of rational functions of two variables with real coefficients. Each rational function ▫$f \in {\mathcal F}$▫ can be thought as a continuous function ▫$f ... \colon {\text dom}(f) \to \bar{\mathbb R}$▫ taking values in the projective line ▫$\bar{\mathbb R} = {\mathbb R} \cup \{\infty\}$▫ and defined on a cofinite subset ▫$\text{dom}(f)$▫ of the torus ▫$\bar{\mathbb R}^2$▫. Then the family ▫${\mathcal F}$▫ determines a continuous vector-function ▫$\mathcal{F} \colon {\text dom}(\mathcal{F}) \to \bar{\mathbb R}^\mathcal{F}$▫ defined on the dense ▫$G_\delta$▫-set ▫$\text{dom}(\mathcal{F}) = \bigcap_{f\in \mathcal{F}}\text{dom}(\mathcal{F})$▫ of ▫$\bar{\mathbb R}^2$▫. The closure ▫$\bar{\Gamma} (\mathcal{F})$▫ of its graph ▫$\Gamma(\mathcal{F}) = \{(x,f(x)): \; x \in \text{dom}(\mathcal{F})\}$▫ in ▫$\bar{\mathbb R}^2 \times \bar{\mathbb R}^\mathcal{F}$▫ is called the graphoid of the family ▫${\mathcal F}$▫. We prove the graphoid ▫$\bar{\Gamma} (\mathcal{F})$▫ has topological dimension ▫$\dim(\bar{\Gamma} (\mathcal{F})) = 2$▫. If the family ▫${\mathcal F}$▫ contains all linear fractional transformations ▫$f(x,y) = \frac{x-a}{y-b}$▫ for ▫$(a,b) \in {\mathbb Q}^2$▫, then the graphoid ▫$\bar{\Gamma} (\mathcal{F})$▫ has cohomological dimension ▫$\dim_G(\bar{\Gamma} (\mathcal{F})) = 1$▫ for any non-trivial 2-divisible abelian group ▫$G$▫. Hence the space ▫$\bar{\Gamma} (\mathcal{F})$▫ is a natural example of a compact space that is not dimensionally full-valued and by this property resembles the famous Pontryagin surface.
    Vir: Topology and its Applications. - ISSN 0166-8641 (Vol. 160, iss. 1, 2013, str. 24-44)
    Vrsta gradiva - članek, sestavni del
    Leto - 2013
    Jezik - angleški
    COBISS.SI-ID - 16486745

    Povezava(-e):

    http://dx.doi.org/10.1016/j.topol.2012.09.012

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