# ωω-dominated function spaces and ωω-bases in free objects of topological algebra Academic Article

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### abstract

• 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′) .

### publication date

• January 1, 2018