The set of space-filling curves: topological and algebraic structure

TL;DR

This paper investigates the topological and algebraic properties of families of space-filling curves, including Peano curves and those with positive Jordan content, focusing on their size and structure.

Contribution

It provides a detailed analysis of the topological and algebraic structures of space-filling curves, highlighting their size and properties within these frameworks.

Findings

Analysis of the size of families of space-filling curves

Topological and algebraic structure characterization

Comparison between Peano and general space-filling curves

Abstract

In this paper, a study of topological and algebraic properties of two families of functions from the unit interval $I$ into the plane $\mathbb{R}^2$ is performed. The first family is the collection of all Peano curves, that is, of those continuous mappings onto the unit square. The second one is the bigger set of all space-filling curves, i.e. of those continuous functions $I \to \mathbb{R}^2$ whose images have positive Jordan content. Emphasis is put on the size of these families, in both topological and algebraic senses, when endowed with natural structures.