## Abstract

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∈ωω } 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∈ωω } 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^{♯}=ω_{1} (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 |
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Pages (from-to) | 203-241 |

Number of pages | 39 |

Journal | Topology and its Applications |

Volume | 241 |

DOIs | |

State | 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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