  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.
