Peano curves on topological vector spaces

TL;DR

This paper explores the existence and algebraic structure of Peano curves in topological vector spaces, demonstrating large algebras within these families and analyzing their properties as continuous images of the real line.

Contribution

It introduces new results on the lineability of families of Peano curves in topological vector spaces and characterizes these spaces as continuous images of the real line.

Findings

Large algebras of Peano curves are constructed.

Topological vector spaces that are continuous images of the real line are characterized.

Optimal lineability results are established for these spaces.

Abstract

The starting point of this paper is the existence of Peano curves, that is, continuous surjections mapping the unit interval onto the unit square. From this fact one can easily construct of a continuous surjection from the real line $\mathbb{R}$ to any Euclidean space $\mathbb{R}^n$. The algebraic structure of the set of these functions (as well as extensions to spaces with higher dimensions) is analyzed from the modern point of view of lineability, and large algebras are found within the families studied. We also investigate topological vector spaces that are continuous image of the real line, providing an optimal lineability result.