Construction of ball spaces and the notion of continuity

TL;DR

This paper develops methods for constructing new spherically complete ball spaces from existing ones, introduces a notion of continuity, and explores their categorical properties to facilitate fixed point theorem applications.

Contribution

It introduces set-theoretic constructions of new ball spaces, defines continuity and quotient spaces, and establishes a topological category framework for ball spaces.

Findings

Construction methods for new ball spaces via set operations

Definition of continuity and quotient spaces in ball spaces

Existence of products and coproducts in the category of ball spaces

Abstract

Spherically complete ball spaces provide a framework for the proof of generic fixed point theorems. For the purpose of their application it is important to have methods for the construction of new spherically complete ball spaces from given ones. Given various ball spaces on the same underlying set, we discuss the construction of new ball spaces through set theoretic operations on the balls. A definition of continuity for functions on ball spaces leads to the notion of quotient spaces. Further, we show the existence of products and coproducts and use this to derive a topological category associated with ball spaces.