Jordan and Schoenflies in non-metrical analysis situs

TL;DR

This paper extends the classical Jordan and Schoenflies theorems to non-metric 2-dimensional manifolds and explores their implications in dynamical systems, broadening the scope of topological analysis.

Contribution

It provides the first extension of these fundamental theorems to non-metric manifolds in the two-dimensional case, with applications to dynamical systems.

Findings

Jordan and Schoenflies theorems hold in non-metric 2-manifolds

Applications to Poincaré-Bendixson theory are demonstrated

Results expand topological tools for non-metric spaces

Abstract

We show that both, the Jordan curve theorem and the Schoenflies theorem extend to non-metric manifolds (at least in the two-dimensional context), and conclude by some dynamical applications \`a la Poincar\'e-Bendixson.