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

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14 Scopus citations

## 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 e1 (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).

Original language English 203-241 39 Topology and its Applications 241 https://doi.org/10.1016/j.topol.2018.03.021 Published - 1 Jun 2018

## Keywords

• Free Abelian topological group
• Free Boolean topological group
• Free linear topological space
• Free locally convex space
• Free topological group
• Monotone cofinal map
• Uniform space
• ω-base

## ASJC Scopus subject areas

• Geometry and Topology

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