A Jordan Curve Theorem for 2-dimensional Tilings

TL;DR

This paper extends the Jordan curve theorem to all locally finite tilings of the plane, generalizing classical results and including the Khalimsky plane as a special case, thus broadening the theorem's applicability.

Contribution

It proves a Jordan curve theorem for any locally finite tiling of the plane, generalizing previous results for specific grids and the Khalimsky plane.

Findings

Validates the Jordan curve theorem for all locally finite tilings

Generalizes Rosenfeld's theorem for point grids

Includes the Khalimsky plane as a special case

Abstract

The classical Jordan curve theorem for digital curves asserts that the Jordan curve theorem remains valid in the Khalimsky plane. Since the Khalimsky plane is a quotient space of $\mathbb R^2$ induced by a tiling of squares, it is natural to ask for which other tilings of the plane it is possible to obtain a similar result. In this paper we prove a Jordan curve theorem which is valid for every locally finite tiling of $\mathbb R^2$. As a corollary of our result, we generalize some classical Jordan curve theorems for grids of points, including Rosenfeld's theorem.