  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 $U_\beta[x]\subset
U_\alpha[x]$ for all $\alpha\le\beta$ 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.
