Dimension of graphoids of rational vector-functions

作者：Banakh Taras^*; Potyatynyk Oles

来源：Topology and Its Applications, 2013, 160(1): 24-44.

DOI：10.1016/j.topol.2012.09.012

• Graphoid
• Graph
• Rational vector-function
• Topological dimension
• Extension dimension
• Cohomological dimension
• Pontryagin surface

Let F subset of R(x. y) be a countable family of rational functions of two variables with real coefficients. Each rational function f is an element of F can be thought as a continuous function f : dom(f) -> (R) over bar taking values in the projective line (R) over bar = R boolean OR {infinity} and defined on a cofinite subset dom(f) of the torus (R) over bar (2). Then the family F determines a continuous vector-function F : dom(F) -> (R) over bar (F) defined on the dense G(delta)-set dom(F) = boolean AND(f is an element of F) dom(F) of (R) over bar (2). The closure (Gamma) over bar (F) of its graph Gamma(F) = {(x. f(x)): x is an element of dom(F)} in (R) over bar (2) x (R) over bar (F) is called the graphoid of the family F. We prove that the graphoid (Gamma) over bar (F) has topological dimension dim((Gamma) over bar (F)) = 2. If the family F contains all linear fractional transformations f (x. y) = x-a/y-b (a, b) is an element of Q(2), then the graphoid (Gamma) over bar (F) has cohomological dimension dim(G)((Gamma) over barF)) = 1 for any non-trivial 2-divisible abelian group G. Hence the space (Gamma) over bar (F) is a natural example of a compact space that is not dimensionally full-valued and by this property resembles the famous Pontryagin surface.

• 出版日期2013-1-1