Darboux homotopies and Darboux retracts -- Results and questions (Q1898987)

In this paper the notions of \textit{Darboux homotopy} and \textit{Darboux retract} are defined and used for the description of some topological objects, e.g., complete rings of Darboux functions mapping \([0, 1]\) into \(\mathbb{R}\) (recall that a ring \(\mathcal P\) is complete if \(|f|\in {\mathcal P}\) for each \(f\in {\mathcal P}\)) or components of a metric separable space. Moreover, the paper contains a lot of interesting open questions concerning these problems.