Dense families of selections and finite-dimensional spaces

TL;DR

This paper characterizes finite-dimensional spaces using continuous selections that avoid certain sets, and extends selection theorems to countable-dimensional spaces, leading to new proofs of classical theorems in topology.

Contribution

It introduces a new characterization of finite-dimensional spaces via selection theory and generalizes selection theorems to strongly countable-dimensional spaces.

Findings

Characterization of n-dimensional spaces via continuous selections avoiding Z_n-sets

A selection theorem for strongly countable-dimensional spaces

New proof of the Hurewicz formula

Abstract

A characterization of $n$-dimensional spaces via continuous selections avoiding $Z_n$-sets is given, and a selection theorem for strongly countable-dimensional spaces is established. We apply these results to prove a generalized Ostrand's theorem and to obtain a new alternative proof of the Hurewicz formula.