Countable tightness and $${\mathfrak {G}}$$-bases on free topological groups

Abstract

Given a Tychonoff space X, let F (X) and A(X) be respectively the free topological group and the free Abelian topological group over X in the sense of Markov. In this paper, we consider two topological properties of F (X) or A(X), namely the countable tightness and G-base. We provide some characterizations of the countable tightness and G-base of F (X) and A(X) for various special classes of spaces X. Furthermore, we also study the countable tightness and G-base of some Fn(X) of F (X).2000 Mathematics Subject Classification. Primary 54H11, 22A05; Secondary 54E20; 54E35; 54D50; 54D55.

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

“…Remark 1.3. Theorem 1.2 completely resolves the problem of characterization of Tychonoff spaces X whose free objects A(X), F(X), L(X) have G-bases, posed in [23,Question 4.15] and then repeated (and partially answered) in [20, §3], [21], [30], [31]. Similar problems are also considered and (partly) resolved in [25, §5.2 and §6.5].…”

“…For the first time the concept of an ω ω -base appeared in [16] as a tool for studying locally convex spaces that belong to the class G introduced by Cascales and Orihuela [11]. A systematic study of locally convex spaces and topological groups with an ω ω -base was started in [22], [23] and continued in [18], [20], [31]. In these papers ω ω -bases are called G-bases, but we prefer to use the terminology of ω ω -bases, which is more self-suggesting and flexible.…”

“…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].…”

“…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].…”

A classification of the cofinal structures of precompacta

“…For functions f, g ∈ N N , define f ≤ g if f (n) ≤ g(n) for all natural numbers n. For a topological group G, let N (G) be the family of neighborhoods of the identity element in G. An N N -base (also known as G-base) of a topological group G is the image of a a monotone cofinal map from (N N , ≤) to (N (G), ⊇). The notion of N Nbase has recently attracted considerable attention [3,4,8,9,10,14,15]. Gabriyelyan, Kąkol, and Leiderman [9] apply Christensen's result to prove that, for Polish spaces X, the topological groups C(X, R) have N N -bases.…”