Preserving $Z$-sets by Dranishnikov's resolution

Abstract

We prove that Dranishnikov's $k$-dimensional resolution $d_k\colon μ^k\to Q$ is a UV$^{n-1}$-divider of Chigogidze's $k$-dimensional resolution $c_k$. This fact implies that $d_k^{-1}$ preserves $Z$-sets. A further development of the concept of UV$^{n-1}$-dividers permits us to find sufficient conditions for $d_k^{-1}(A)$ to be homeomorphic to the Nöbeling space $ν^k$ or the universal pseudoboundary $σ^k$. We also obtain some other applications.