Finite dimensions modulo simplicial complexes and $ANR$-compacta

Abstract

New dimension functions $\Cal{G}$-dim and$\Cal{R}$-dim, where $\Cal{G}$ is a class of finitesimplicial complexes and $\Cal{R}$ is a class of $ANR$-compacta, areintroduced. Their definitions are based on the theorem on partitionsand on the theorem on inessential mappings to cubes, respectively.If $\Cal{R}$ is a class of compact polyhedra, then for its arbitrarytriangulation $\tau$, we have ${\Cal{R}}_\tau\text{-dim}\,X={\Cal{R}}\text{-dim}\,X$for an arbitrary normal space $X$. To investigate thedimension function $\Cal{R}$-dim we apply results ofextension theory. Internal properties of this dimension function aresimilar to those of the Lebesgue dimension. The following inequality$\Cal{R}\text{-dim}\,X\leq\tx{\rm dim}\,X$holds for an arbitrary class $\Cal{R}$. We discuss the followingQuestion: When $\Cal{R}$-$\text{\rm dim}\,X<ınfty\Rightarrow\text{\rm dim}\,X<ınfty$?