On topological spaces and topological groups with certain local countable networks

Abstract

The strong Pytkeev property Small base Baire space Topological group Function space (Free) locally convex space Free (abelian) topological groupBeing motivated by the study of the space C c (X) of all continuous real-valued functions on a Tychonoff space X with the compact-open topology, we introduced in [16] the concepts of a cp-network and a cn-network (at a point x) in X. In the present paper we describe the topology of X admitting a countable cp-or cn-network at a point x ∈ X. This description applies to provide new results about the strong Pytkeev property, already well recognized and applicable concept originally introduced by Tsaban and Zdomskyy [44]. We show that a Baire topological group G is metrizable if and only if G has the strong Pytkeev property. We prove also that a topological group G has a countable cp-network if and only if G is separable and has a countable cp-network at the unit. As an application we show, among the others, that the space D (Ω) of distributions over open Ω ⊆ R n has a countable cp-network, which essentially improves the well known fact stating that D (Ω) has countable tightness. We show that, if X is an MK ω -space, then the free topological group F (X) and the free locally convex space L(X) have a countable cp-network. We prove that a topological vector space E is p-normed (for some 0 < p ≤ 1) if and only if E is Fréchet-Urysohn and admits a fundamental sequence of bounded sets.

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

“…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].…”

Free locally convex spaces with a small base

“…It is easy to see that every metrizable group has a G-base at the identity. Topological groups with a G-base are thoroughly studied in [19], see also [4,16,18].…”

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

“…Recently, S.S. Gabriyelyan and J. Kakol in the paper [17] and A.G. Leiderman, V.G. Pestov , A.H. Tomita in the paper [26] have given an answer to Questions 1.1 and 1.2 respectively.…”