  We prove that for a complete quasivariety $K$ of topological $E$-algebras of
countable discrete signature $E$ and each submetrizable $ANR(k_\omega)$-space
$X$ its free topological $E$-algebra $F_K(X)$ in the class $K$ is a
submetrizable $ANR(k_\omega)$-space.
