Hyperspaces in topological Categories
Ren\'e Bartsch

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 to a subset of , just to motivate an interesting question in set theory. Then we will propose and discuss a new approach to…
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Taxonomy
TopicsComputability, Logic, AI Algorithms · Advanced Algebra and Logic · Advanced Topology and Set Theory
