Directed and irreversible path in Euclidean spaces

Abstract

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