A combinatorial characterization of Hurewicz cofibrations between finite topological spaces (Q1731349)

In the paper under review the authors provide a characterization of Hurewicz cofibrations between finite topological spaces. Indeed, they show that if \(X\) is a connected finite topological space and \(A\) a non-empty subspace of \(X\), then the inclusion \(i:A\hookrightarrow X\) is a cofibration if, and only if, there exists a retraction \(r:X\rightarrow A\) such that \(ir\leq Id_X.\) In particular, a cofibration between connected non-empty finite topological spaces is a homotopy equivalence. The authors also give an algorithm that detects cofibrations provided we are dealing with continuous maps between finite topological spaces.