ω^ω-dominated function spaces and ω^ω-bases in free objects of topological algebra

A topological space X is defined to have an ω^ω-base if at each point x∈X the space X has a neighborhood base (U_α[x])_α∈ω^ω such that U_β[x]⊂U_α[x] for all α≤β in ω^ω. For a Tychonoff space X consider the following conditions (A) the free Abelian topological group A(X) of X has an ω^ω-base; (B) the free Boolean topological group B(X) of X has an ω^ω-base; (F) the free topological group F(X) of X has an ω^ω-base; (L) the free locally convex space L(X) of X has an ω^ω-base; (V) the free topological vector space V(X) of X has an ω^ω-base; (U) the universal uniformity U_X of X has a base (U_α)_α∈ω^ω such that U_β⊂U_α for all α≤β in ω^ω;(C) the function space C(X) is ω^ω-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^′).