Peano curves on topological vector spaces

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.

💡 Research Summary

The paper investigates the algebraic richness of families of continuous surjections (Peano curves) from the real line (or Euclidean spaces) onto various topological vector spaces. Starting from the classical fact that a continuous surjection exists from the unit interval onto the unit square (the Peano curve), the authors note that this immediately yields continuous surjections from ℝ onto any Euclidean space ℝⁿ. The main goal is to study the linear and algebraic structure of the set of such functions using the modern concepts of lineability, dense‑lineability, spaceability, algebrability, and strong algebrability.

Section 1 reviews previous work on surjective functions (ES, SES, PES, Jones functions) and introduces the necessary terminology. A “Peano space” is defined as a compact, connected, locally connected, second‑countable Hausdorff space, i.e., precisely the spaces that are continuous images of the unit interval by the Hahn‑Mazurkiewicz theorem.

In Section 2 the authors focus on Euclidean targets. Earlier results (Theorems 2.1 and 2.2) established maximal lineability and maximal dense‑lineability of the set C_S(ℝ^m,ℝⁿ) and of C_S^∞(ℝ^m,ℝⁿ) (the latter consisting of surjections whose pre‑image of each point is unbounded). The paper then moves to the complex setting, where the sign restriction of real functions disappears, allowing the construction of large algebras. A key technical tool is Lemma 2.4, which relates the order (growth rate) of entire functions to the order of a polynomial evaluated at several such functions with distinct orders. Using entire functions of non‑integer order, the authors build a free algebra generated by a continuum of functions, each of which is a surjection ℝ→ℂ. Composing these generators with any fixed f∈C_S^∞(ℝ,ℂ) yields a free algebra contained in C_S^∞(ℝ,ℂ). Consequently, Theorem 2.5 asserts that C_S^∞(ℝ^m,ℂⁿ) is maximal strongly algebrable (i.e., strongly c‑algebrable) in C(ℝ^m,ℂⁿ).

Section 3 introduces σ‑Peano spaces: a topological space X that can be written as an increasing union of Peano subspaces. Proposition 3.2 shows that X is σ‑Peano if and only if there exists a continuous surjection ℝ→X (equivalently, a continuous surjection with unbounded fibers). Examples include all Euclidean spaces, Hilbert cubes, and separable dual Banach spaces equipped with the weak* topology. Theorem 3.6 proves that for any σ‑Peano topological vector space X, the set C_S^∞(ℝ^m,X) is c‑lineable, and hence maximal lineable, inside C(ℝ^m,X). This gives an optimal lineability result for continuous images of the line.

Sections 4 and 5 sketch extensions to sequence spaces (ℓ^p, c₀, etc.) and function spaces (C(K), L^p, etc.). The same techniques—using σ‑Peano decompositions and the construction of free algebras via entire functions—apply, showing that analogous maximal lineability and algebrability hold in these infinite‑dimensional contexts.

Overall, the paper blends classical topological results (Hahn‑Mazurkiewicz) with recent developments in lineability theory. It demonstrates that the family of continuous surjections from ℝ (or ℝ^m) onto a wide class of topological vector spaces is not merely large in cardinality but possesses rich linear and algebraic substructures: infinite‑dimensional vector spaces, dense subspaces, closed infinite‑dimensional subspaces, and even free algebras generated by continuum many elements. These findings deepen our understanding of the interplay between topology, functional analysis, and algebraic structure in the realm of space‑filling curves.