### abstract

- A topological space X is defined to have a local G-base if every point x∈ X has a neighborhood base (Uα) α∈ ωω such that Uβ⊂ Uα for all α≤ β in ωω. We prove that for every Tychonoff space X the following conditions are equivalent:(1) the free locally convex space L (X) of X has a local G-base;(2) the free topological vector space V (X) of X has a local G-base;(3) the finest uniformity U (X) of X admits a G-base and the function space C (X) is ωω-dominated. The conditions (1)–(3) imply that every metrizable continuous image of X is σ-compact and all finite powers of X are countably tight. If the space X is separable, then the conditions (1)–(3) imply that X is a countable union of compact metrizable spaces and (1)–(3) are equivalent to:(4) the finest uniformity U (X) of X has a G-base. If the space X is first-countable and perfectly normal, then the conditions (1)–(3) are equivalent to: