A Fixed Point Conjecture

TL;DR

This paper explores a fixed point conjecture linking the existence of fixed points for arbitrary mappings to the non-void nature of certain direct limits of finite partitions.

Contribution

It proposes a novel conjecture connecting fixed point existence with properties of direct limits of partitions, offering new insights into fixed point theory.

Findings

Conjecture relating fixed points to direct limits of partitions

Insights into conditions for fixed point existence

Potential implications for fixed point theorems

Abstract

Inverse limits, unlike direct limits, can in general be void, [1]. The existence of fixed points for arbitrary mappings $T : X \longrightarrow X$ is conjectured to be equivalent with the fact that related direct limits of all finite partitions of X are not void.