Directed and irreversible path in Euclidean spaces

Abstract

The aim of this very short note is to relate the directed paths in $\stackrel{⟶}{{\mathbb{R}}^{\mathrm{n}}}$ to the irreversible paths in $\stackrel{\mathrm{ir}}{{\mathbb{R}}^{\mathrm{n}}}$. We first show that there is a directed path from $x$ to $y$ in $\stackrel{⟶}{{\mathbb{R}}^{\mathrm{n}}}$ iff there exists an irreversible path with same initial and terminal points in $\stackrel{\mathrm{ir}}{{\mathbb{R}}^{\mathrm{n}}}$. Also, we prove that every directed path in $\stackrel{⟶}{{\mathbb{R}}^{\mathrm{n}}}$ is an irreversible path in $\stackrel{\mathrm{ir}}{{\mathbb{R}}^{\mathrm{n}}}$.

References in corpus (2)

• On directed homotopy equivalences and a notion of directed topological complexity
• Irreversible homotopy and a notion of irreversible Lusternik-Schnirelmann category