${\omega }^{\omega }$-Dominated function spaces and ${\omega }^{\omega }$-bases in free objects of Topological Algebra

A topological space $X$ is defined to have an ${\omega }^{\omega }$-base if at each point $x\in X$ the space $X$ has a neighborhood base $(U_{\alpha }[x]{)}_{\alpha \in {\omega }^{\omega }}$ such that $U_\beta[x]\subset U_\alpha[x]$forall$\alpha\le\beta$in$\omega^\omega$. We characterize topological and uniform spaces whose free (locally convex) topological vector spaces or free (Abelian or Boolean) topological groups have ${\omega }^{\omega }$-bases.