Topological Expansion, Study and Applications

TL;DR

This paper introduces the concept of topological expansion using local multi-homeomorphisms, demonstrating that fractal families of topological spaces can be expanded, leading to finer topologies and applications in fractal manifolds.

Contribution

It defines topological expansion via coproduct topology and proves that fractal families are expanding, establishing a foundation for locally expandable and fractal manifold topologies.

Findings

Fractal families of topological spaces are expanding.

Expanding spaces have finer topologies.

Fractal manifolds are locally expandable and have natural topological expansions.

Abstract

In this paper, we introduce the notion of expanding topological space. We define the topological expansion of a topological space via local multi-homeomorphism over coproduct topology, and we prove that the coproduct family associated to any fractal family of topological spaces is expanding. In particular, we prove that the more a topological space expands, the finer the topology of its indexed states is. Using multi-homeomorphisms over associated coproduct topological spaces, we define a locally expandable topological space and we prove that a locally expandable topological space has a topological expansion. Specifically, we prove that the expanding fractal manifold is locally expandable and has a natural topological expansion.