### abstract

- Abstract: 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.