A study of topological structures on equi-continuous mappings

TL;DR

This paper develops and characterizes topological structures on the space of equi-continuous mappings, introducing properties like splittingness and admissibility, and analyzing their dual topologies.

Contribution

It introduces new topological properties for equi-continuous function spaces and characterizes their dualities, advancing understanding of their structural features.

Findings

Open-entourage topology is admissible.

Point-transitive-entourage topology is splitting.

Dual topologies preserve admissibility and splittingness.

Abstract

Function space topologies are developed for EC(Y,Z), the class of equi-continuous mappings from a topological space Y to a uniform space Z. Properties such as splittingness, admissibility etc. are defined for such spaces. The net theoretic investigations are carried out to provide characterizations of splittingness and admissibility of function spaces on EC(Y,Z). The open-entourage topology and point-transitive-entourage topology are shown to be admissible and splitting respectively. Dual topologies are defined. A topology on EC(Y,Z) is found to be admissible (resp. splitting) if and only if its dual is so.