Remarks on Fixed Point Assertions in Digital Topology

TL;DR

This paper critically examines fixed point theorems in digital topology, highlighting errors and trivial results arising from the inappropriate adaptation of classical topological tools.

Contribution

It reveals common mistakes in applying classical topology methods to digital topology, emphasizing the need for more suitable approaches.

Findings

Many fixed point results are incorrect or trivial when classical tools are misapplied.

The paper identifies specific issues with the use of metric spaces and homotopy in digital topology.

It advocates for developing methods tailored to digital topology rather than relying on classical analogs.

Abstract

Several recent papers in digital topology have sought to obtain fixed point results by mimicking the use of tools from classical topology, such as complete metric spaces and homotopy invariant fixed point theory. We show that in many cases, researchers using these tools have derived conclusions that are incorrect or trivial.