Fractal Topology Foundations

TL;DR

This paper develops a foundational framework for fractal topological spaces using nested topologies, illustrating how increasing complexity yields stronger structures, and introduces a fractal manifold model as an example.

Contribution

It presents a novel construction of fractal topological spaces through nested topologies and introduces a fractal manifold model as an application.

Findings

Nested topologies create increasingly complex structures.

The fractal manifold is locally homeomorphic to the fractal topological space.

The framework links the number of spaces to structural strength.

Abstract

In this paper, we introduce the foundation of a fractal topological space constructed via a family of nested topological spaces endowed with subspace topologies, where the number of topological spaces involved in this family is related to the appearance of new structures on it. The greater the number of topological spaces we use, the stronger the subspace topologies we obtain. The fractal manifold model is brought up as an illustration of space that is locally homeomorphic to the fractal topological space.