Free locally convex spaces with a small base

Abstract

Abstract. The paper studies the free locally convex space L(X) over a Tychonoff space X. Since for infinite X the space L(X) is never metrizable (even not Fréchet-Urysohn), a possible applicable generalized metric property for L(X) is welcome. We propose a concept (essentially weaker than first-countability) which is known under the name a G-base. A space X has a G-base if for every x ∈ X there is a base {Uα : α ∈ N N } of neighborhoods at x such that U β ⊆ Uα whenever α ≤ β for all α, β ∈ N N , where α = (α(n)) n∈N ≤ β = (β(n)) n∈N if α(n) ≤ β(n) for all n ∈ N. We show that if X is an Ascoli σ-compact space, then L(X) has a G-base if and only if X admits an Ascoli uniformity U with a G-base. We prove that if X is a σ-compact Ascoli space of N N -uniformly compact type, then L(X) has a G-base. As an application we show: (1) if X is a metrizable space, then L(X) has a G-base if and only if X is σ-compact, and (2) if X is a countable Ascoli space, then L(X) has a G-base if and only if X has a G-base.

Bounded sets structure of CpX and quasi‐(DF)‐spaces

“…By Ascoli's theorem , each k ‐space is Ascoli. The following proposition complements [, Theorem 3.2] and [, Theorem 6.5]. In fact the equivalence between (i) and (iii) below has been proved in [, Theorem 6.5] (see also [, Theorem 1.2]).…”

“…Therefore is -compact by Corollary 9.2 of [30]. Conversely, if is -compact, then ( ) has a fundamental compact resolution by Corollary 2.10 of [22]. Once again applying Theorem 2 of [16], we obtain that has a -base.…”

“…The equivalence (iii)⇔(iv) is Corollary 9.2 of , and the implication (i)⇒(iii) is trivial. The implication (iv)⇒(i) follows from Corollary 2.10 of which states that has even a fundamental compact resolution.…”

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

“…Free topological vector spaces V(X) over Tychonoff spaces are introduced and studied in a recent preprint [24], but with completely different approach and results. Note also that several results about the existence of ω ω -bases in free locally convex spaces were proved independently (and using different arguments than ours) in [21]. In particular, the authors of [21] proved that a metrizable space X is σ-compact if and only if its free locally convex space L(X) has an ω ω -base; also they proved that for a countable Ascoli space X with an ω ω -base, the free locally convex space L(X) has an ω ω -base.…”

“…It was shown earlier that (a) for a cosmic k ω -space the free objects A(X), F(X), L(X) have ω ω -bases [23], [20], [21], [31]. (b) for a metrizable σ-compact space X the free locally convex space L(X) has an ω ω -base [21]; 1 (c) the free Abelian topological group A(X) of a Tychonoff space X has an ω ω -base if and only if the universal uniformity U X of X has an ω ω -base [30], [32], [25]; (d) a uniform space X is ω ω -based iff the free Abelian topological group A u (X) has an ω ω -base [30]; (e) a separable uniform space X is ω ω -based iff the free topological group F u (X) has an ω ω -base [30]. In this paper we shall characterize uniform spaces whose free topological groups and free (locally convex) topological vector spaces have ω ω -bases.…”