On closed embeddings of free topological algebras

TL;DR

This paper proves that for certain classes of topological algebras, the natural homomorphism from the free algebra over a closed subspace to the free algebra over the larger space is a closed embedding, generalizing a known result for free topological groups.

Contribution

It extends the understanding of embeddings of free topological algebras to a broader class of algebraic structures and spaces, using extension properties of the Hartman-Mycielski construction.

Findings

Homomorphism from F(X) to F(Y) is a closed topological embedding for closed X in metrizable or stratifiable Y.

Generalizes Uspenskii's result on free topological groups.

Uses extension properties of the Hartman-Mycielski construction.

Abstract

Let $\mathcal K$ be a complete quasivariety of completely regular universal topological algebras of continuous signature $\mathcal E$ (which means that $\mathcal K$ is closed under taking subalgebras, Cartesian products, and includes all completely regular topological $\mathcal E$-algebras algebraically isomorphic to members of $\mathcal K$). For a topological space $X$ by $F(X)$ we denote the free universal $\mathcal E$-algebra over $X$ in the class $\mathcal K$. Using some extension properties of the Hartman-Mycielski construction we prove that for a closed subspace $X$ of a metrizable (more generally, stratifiable) space $Y$ the induced homomorphism $F(X)\to F(Y)$ between the respective free universal algebras is a closed topological embedding. This generalizes one result of V.Uspenskii concerning embeddings of free topological groups.