Dugundji's Canonical Covers, asymptotic and covering dimension

TL;DR

This paper characterizes the dimension of certain closed subsets in compact metrizable spaces using canonical covers, especially for Z-sets in Hilbert and finite-dimensional cubes, addressing open questions in the field.

Contribution

It provides a new characterization of dimension via canonical covers and solves existing open questions related to Z-sets in specific compact spaces.

Findings

Dimension of X characterized by multiplicity of canonical covers

Results applied to Z-sets in Hilbert cube and finite-dimensional cube

Addresses and resolves open questions in the literature

Abstract

Given a nowheredense closed subset $X$ of a metrizable compact space $\tx$, we characterize the dimension of $X$ in terms of the multiplicity of the canonicals covers of the complementary of $X$, specially in some particular cases, like when $\tx$ is the Hilbert cube or the finite dimensional cube and $X$, a Z-set of $\tx$. In this process, we solve some questions in the literature.