A combinatorial characterization of Hurewicz cofibrations between finite topological spaces

TL;DR

This paper provides a complete combinatorial characterization of Hurewicz cofibrations between finite topological spaces, revealing that such cofibrations between connected non-empty spaces are homotopy equivalences and offering an algorithm for their detection.

Contribution

It introduces a novel combinatorial framework for identifying Hurewicz cofibrations in finite topological spaces and presents an efficient algorithm for their recognition.

Findings

Cofibrations between connected non-empty finite spaces are homotopy equivalences.

A simple algorithm can determine whether a function is a cofibration.

The characterization simplifies understanding of homotopy properties in finite spaces.

Abstract

We characterize the Hurewicz cofibrations between finite topological spaces, that is, the continuous functions between finite topological spaces that have the homotopy extension property with respect to all topological spaces. In particular, we show that cofibrations between connected non-empty finite topological spaces are homotopy equivalences. As a consequence of our characterization, we obtain a simple algorithm capable of determining whether a given continuous function between finite topological spaces is a cofibration.