The Planck boundary within the hyperspace of the circle of pseudo-arcs

TL;DR

This paper explores a unique geometric structure in hyperspaces of certain continua, revealing a complex interplay between smooth and non-smooth partitions that parallels macroscopic and quantum universe realms.

Contribution

It introduces a novel geometric perspective on hyperspaces of decomposable, non-locally connected homogeneous continua, highlighting the Planck boundary within this context.

Findings

Identification of nonnegative metric curvature in hyperspaces

Discovery of the coexistence of smooth and non-smooth partitions

Analogy between geometric structures and universe realms

Abstract

In this paper we point out an interesting geometric structure of nonnegative metric curvature emerging from the hyperspaces of decomposable, non-locally connected homogeneous continua, where "smooth" and "non-smooth" partitions live together, similarly to the macroscopic and the quantum realms of the Universe.