Mazurkiewicz Manifolds and Homogeneity

It is proved that no region of a homogeneous locally compact, locally connected metric space can be cut by an Fσ-subset of a “smaller” dimension. The result applies to different finite or infinite topological dimensions of metrizable spaces. The classical Hurewicz-Menger-Tumarkin theorem in dimension theory says that connected topological n-manifolds (with or without boundary) are Cantor manifolds (i. e., no subset of covering dimension ≤ n−2 separates the space). The theorem was almost immediately strengthened by Mazurkiewicz who proved that regions (i. e., open connected subsets) in Euclidean spaces (and, in fact, in topological manifolds) cannot be cut by subsets of codimension at least two (a subset cuts if its complement is not continuum-wise connected [2]). The HurewiczMenger-Tumarkin theorem has many generalizations. In particular, it is known that regions of homogeneous locally compact metric spaces are Cantor manifolds (including their infinite-dimensional versions) [5, 6]. It was proved in [3] that no weakly infinite-dimensional subset cuts the product of a countable number of nondegenerate metric continua. In this paper, we obtain a generalization in spirit of the Mazurkiewicz theorem: regions in homogeneous locally compact, locally connected metric spaces cannot be cut by Fσ-subsets of codimension at least two. Moreover, our result holds true for a very general dimension function DK considered in [4] which captures the covering dimension, cohomological dimension dimG with respect to any Abelian group G as well as the extraordinary dimension dimL with respect to a given CW complex L, and has its counterparts in infinite dimensions including C-spaces and weakly infinite-dimensional spaces. Basic facts on Cantor manifolds and their stronger variations with respect to dimension DK or to the above-mentioned infinite dimensions Date: January 9, 2009. 2000 Mathematics Subject Classification. Primary 54F45; Secondary 55M10.