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.
Categories: math.GN