Fixed Point Results in General Topology Continuous Mappings and Separation Axioms

Abstract

The principles underlying the fixed point theory are being methodically explored in this research paper, in the abstract framework of general topology, and no longer in the more traditional distance-based restrictions of the metric spaces. The paper is dedicated to the severe interaction between continuous mappings and separation axioms in the determination of the Fixed Point Property (FPP) in different topological spaces. We consider the way continuous functions and topological retracts conserve key structural properties, e.g. compactness and connectedness, to ensure the existence of fixed points that meet the condition:

Besides, the article offers a mathematically rigorous appraisal of axioms of separation, showing that the Hausdorff () axiom is a strict requirement in the way of making sure that the fixed-point of a continuous map constitutes a topologically closed set. Through a generalization of the use of classical theorems, such as those of Brouwer, Schauder, and Kakutani, we demonstrate how compactness, convexity and topological resolution are used to give convergence in a single valued and a multi-valued mapping. Lastly, we provide certain counterexamples (e.g. spaces with the cofinite topology and non-compact continuous domains) to illuminate the mathematical pathologies and lack of convergence which occur when these topological axiomatic foundations are weakened or eliminated.