A topological space X is defined to have an ωω-base if at each point x∈X the space X has a neighborhood base (Uα[x])α∈ωω such that Uβ[x]⊂Uα[x] for all α≤β in ωω.For a Tychonoff space X consider the following conditions:(A)the free Abelian topological group A(X) of X has an ωω-base;(B)the free Boolean topological group B(X) of X has an ωω-base;(F)the free topological group F(X) of X has an ωω-base;(L)the free locally convex space L(X) of X has an ωω-base;(V)the free topological vector space V(X) of X has an ωω-base;(U)the universal uniformity UX of X has a base (Uα)α∈ωω such that Uβ⊂Uα for all α≤β in ωω;(C)the function space C(X) is ωω-dominated;(σ)X is σ-compact;(σ′)the set X′ of non-isolated points in X is σ-compact;(s)the space X is separable;(S)X is separable or cov♯(X)≤add(X);(D)X is discrete. Then (L)⇔(V)⇔(U∧C)⇔(U∧σ)⇔(U∧s)⇒(U∧S)⇒(F)⇒(A)⇔(B)⇔(U) and moreover (U∧S)⇔(F) under the set-theoretic assumption e♯=ω1 (which is weaker than b=d).If X is not a P-space, then (L)⇔(V)⇔(U∧C)⇔(U∧σ)⇔(U∧s)⇔(F)⇒(A)⇔(B)⇔(U).If the space X is metrizable, then (L)⇔(V)⇔(σ)⇒(D∨σ)⇔(F)⇒(A)⇔(B)⇔(σ′).