# Constructing Selections Stepwise Over Cones of Simplicial Complexes

## TL;DR

This paper generalizes selection characterizations of paracompact C-spaces and finite C-spaces, providing simplified proofs and new characterizations for finite-dimensional paracompact spaces using simplicial complexes.

## Contribution

It introduces a natural generalization of Uspenskij's selection characterization and applies it to simplify existing proofs and extend characterizations to finite-dimensional paracompact spaces.

## Key findings

- Generalized selection characterizations for paracompact C-spaces
- Simplified proof of Valov's characterization of finite C-spaces
- New characterization of finite-dimensional paracompact spaces

## Abstract

It is obtained a natural generalisation of Uspenskij's selection characterisation of paracompact $C$-spaces. The method developed to achieve this result is also applied to give a simplified proof of a similar characterisation of paracompact finite $C$-space obtained previously by Valov. Another application is a characterisation of finite-dimensional paracompact spaces which generalises both a remark done by Michael and a result obtained by the author.