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 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 → R 2 whose images have the positive Jordan content. Emphasis is put on the size of these families, in both topological and algebraic senses, when endowed with natural structures.