Hyperspaces in topological Categories

TL;DR

This paper introduces a new approach to defining hyperspaces in all cartesian closed topological categories, expanding their applicability and connecting hyperspace theory with broader topological and set-theoretic concepts.

Contribution

It proposes a universal method for constructing hyperspaces across all cartesian closed topological categories, addressing a gap in the existing theory.

Findings

New approach to hyperstructure construction in topological categories

Connections established between hyperspaces and function spaces

Potential applications in set theory and formal language theory

Abstract

Hyperspaces form a powerful tool in some branches of mathematics: lots of fractal and other geometric objects can be viewed as fixed points of some functions in suitable hyperspaces - as well as interesting classes of formal languages in theoretical computer sciences, for example (to illustrate the wide scope of this concept). Moreover, there are many connections between hyperspaces and function spaces in topology. Thus results from hyperspaces help to get new results in function spaces and vice versa. Unfortunately, there ist no natural hyperspace construction known for general topological categories (in contrast to the situation for function spaces). We will shortly present a rather combinatorial idea for the transfer of structure from a set $X$ to a subset of $\mathfrak{P}(X)$, just to motivate an interesting question in set theory. Then we will propose and discuss a new approach to define hyperstructures, which works in every cartesian closed topological category, and so applies to every topological category, using it's topological universe hull.