In this article it is proved that for any quasimetric d on a set X with a base-point p_x there exists a maximal invariant extension p on the free monoid F^a(X, V) in a non-Burnside quasi-variety V of topological monoids (Theorem 6.1). This fact permits to prove that for any non-Burnside quasi-variety V of topological monoids and any T₀-space X the free topological monoid F(X,V) exists and is abstract free (Theorem 8.1). Corollary 10.2 affirms that F(X, V), where V is a non-trivial complete non-Burnside quasi-variety of topological monoids, is a topological digital space if and only if X is a topological digital space.