The free abelian topological group and the free locally convex space on the unit interval

TL;DR

This paper characterizes when free abelian topological groups and free locally convex spaces on a space X embed into those on the unit interval, revealing embedding conditions for various classes of spaces.

Contribution

It provides a complete description of spaces X for which the free abelian topological group A(X) embeds into A(I), and explores similar embedding results for free locally convex spaces.

Findings

A(X) embeds into A(I) iff X is finite-dimensional and compact metrizable.

Any finite-dimensional compact metrizable space X has A(X) embedding into A(I).

Results utilize Kolmogorov's Superposition Theorem.

Abstract

We give a complete description of the topological spaces $X$ such that the free abelian topological group $A(X)$ embeds into the free abelian topological group $A(I)$ of the closed unit interval. In particular, the free abelian topological group $A(X)$ of any finite-dimensional compact metrizable space $X$ embeds into $A(I)$. The situation turns out to be somewhat different for free locally convex spaces. Some results for the spaces of continuous functions with the pointwise topology are also obtained. Proofs are based on the classical Kolmogorov's Superposition Theorem.