Generalized Hyperspaces and Non-Metrizable Fractals

TL;DR

This paper extends the theory of fractals to non-metrizable spaces called $eta$-spaces, generalizing compactness and fixed point concepts to enable fractal construction beyond traditional metric spaces.

Contribution

It introduces a generalized framework for compact sets and continuous functions in $eta$-spaces, allowing the definition of fractals as fixed points of IFS in non-metrizable contexts.

Findings

Generalization of compactness and fixed point theory to $eta$-spaces

Construction of non-metrizable fractals as IFS fixed points

Examples illustrating the new fractal types in non-metrizable spaces

Abstract

Much of the structure in metric spaces that allows for the creation of fractals exists in more generalized non-metrizable spaces. In particular the same theorems regarding the behavior of compact sets can be proven in the more general framework of $\beta$-spaces. However in most $\beta$-spaces, a set being compact (and more generally being totally bounded) is so restrictive as to render all fractal examples completely uninteresting. In this paper we provide a generalization of compact sets, continuous functions, and all the related machinery necessary for fractals to be defined as the unique fixed set of an IFS. We conclude by discussing some interesting examples of non-metrizable fractals.