Preserving $Z$-sets by Dranishnikov's resolution

TL;DR

The paper demonstrates that Dranishnikov's resolution preserves Z-sets and explores conditions under which inverse images are homeomorphic to known universal spaces, advancing the understanding of resolution properties in topology.

Contribution

It establishes that Dranishnikov's resolution acts as a UV^{n-1}-divider, ensuring Z-set preservation, and develops criteria for inverse images to be homeomorphic to universal spaces.

Findings

d_k^{-1} preserves Z-sets

Conditions for inverse images to be homeomorphic to eling space or pseudoboundary

Applications to resolution theory in topology

Abstract

We prove that Dranishnikov's $k$-dimensional resolution $d_k\colon \mu^k\to Q$ 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 N\"{o}beling space $\nu^k$ or the universal pseudoboundary $\sigma^k$. We also obtain some other applications.