On closed embeddings of free topological algebras

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.

3 pages