FINITE DIMENSIONS MODULO SIMPLICIAL COMPLEXES AND ANR-COMPACTA

Abstract. New dimension functions G-dim and R-dim, where G is a class of finite simplicial complexes and R is a class of ANR-compacta, are introduced. Their definitions are based on the theorem on partitions and on the theorem on inessential mappings to cubes, respectively. If R is a class of compact polyhedra, then for its arbitrary triangulation τ, we have Rτ- dimX = R- dimX for an arbitrary normal space X. To investigate the dimension function R-dim we apply results of extension theory. Internal properties of this dimension function are similar to those of the Lebesgue dimension. The following inequality R- dimX ≤ dimX holds for an arbitrary class R. We discuss the following Question: When R-dimX <∞ ⇒ dimX <∞