Finite dimensions modulo simplicial complexes and ANR-compacta

Abstract

New dimension functions $\mathcal{G}$-dim and$\mathcal{R}$-dim, where $\mathcal{G}$ is a class of finitesimplicial complexes and $\mathcal{R}$ 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 $\mathcal{R}$ is a class of compact polyhedra, then for its arbitrarytriangulation $\tau$, we have ${\mathcal{R}}_{\tau }\text{-dim}X=\mathcal{R}\text{-dim}X$for an arbitrary normal space $X$. To investigate thedimension function $\mathcal{R}$-dim we apply results ofextension theory. Internal properties of this dimension function aresimilar to those of the Lebesgue dimension. The following inequality$\mathcal{R}\mathcal{\text{-dim}}\mathcal{X}\le \text{tx}\mathrm{d}\mathrm{i}\mathrm{m}\mathcal{X}$holds for an arbitrary class $\mathcal{R}$. We discuss the followingQuestion: When $\mathcal{R}$-$\text{rm dim}X<ınfty\Rightarrow \text{rm dim}X<ınfty$?