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

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.

References in corpus (1)

• Large free linear algebras of real and complex functions

Cited by in corpus (1)

• Algebraic structure of continuous, unbounded and integrable functions