Abstract
Spherically complete ball spaces provide a simple framework for the encoding of completeness properties of various spaces and ordered structures. This allows to prove generic versions of theorems that work with these completeness properties, such as fixed point theorems and related results. For the purpose of applying the generic theorems, it is important to have methods for the construction of new spherically complete ball spaces from existing 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.
Highlights
Ball spaces have been introduced in [6] as a framework for the proof of generic fixed point theorems for functions which in some way are contracting
In analogy to the case of ultrametric spaces, we will call a nonempty collection N of balls in B a nest of balls if it is totally ordered by inclusion
We show the existence of coproducts and prove that coproducts of spherically complete ball spaces are again spherically complete
Summary
Ball spaces have been introduced in [6] as a framework for the proof of generic fixed point theorems for functions which in some way are contracting. Since most of the generic fixed point theorems work for spherically complete ball spaces, it is an important question under which conditions the spaces (Xi, Bi) being spherically complete implies that so is (X, B) (as is the case for products and coproducts). Returning to the case of several ball spaces on a given set X, we will discuss in Section 6.2 the interesting natural example of ordered abelian groups and fields. Take a symmetrically complete ordered group or field G and let B be the set of all convex sets in G that are finite unions of closed bounded intervals and ultrametric balls. Open question: Does the theorem hold if the condition “convex” is removed?
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