Directed and irreversible path in Euclidean spaces

TL;DR

This paper establishes a relationship between directed paths in Euclidean spaces and irreversible paths, showing their equivalence and inclusion properties, thus linking two concepts in geometric path theory.

Contribution

It demonstrates the equivalence of directed and irreversible paths in Euclidean spaces and shows that every directed path is also an irreversible path.

Findings

Directed path from x to y iff an irreversible path exists with same endpoints.

Every directed path is an irreversible path.

The results connect directed and irreversible path concepts 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 ${\stackrel{\rm ir}{\rm \mathbb{R}^n}}$.