The strong Pytkeev property for topological groups and topological vector spaces

Topology and its Applications

Kubzdela

et al. 2015

Self Cite

Elsevier Gabriyelyan, S.; Kakol, JM.; Kubzdela, A.; López Pellicer, M. (2015). On topological properties of Fréchet locally convex spaces. Topology and its Applications. 192 AbstractWe describe the topology of any cosmic space and any ℵ 0 -space in terms of special bases defined by partially ordered sets. Using this description we show that a Baire cosmic group is metrizable. Next, we study those locally convex spaces (lcs) E which under the weak topology σ(E, E ) are ℵ 0 -spaces. For a metrizable and complete lcs E not containing (an isomorphic copy of) 1 and satisfying the Heinrich density condition we prove that (E, σ(E, E )) is an ℵ 0 -space if and only if the strong dual of E is separable. In particular, if a Banach space E does not contain 1 , then (E, σ(E, E )) is an ℵ 0 -space if and only if E is separable. The last part of the paper studies the question: Which spaces (E, σ(E, E )) are ℵ 0 -spaces? We extend, among the others, Michael's results by showing: If E is a metrizable lcs or a (DF )-space whose strong dual E is separable, then (E, σ(E, E )) is an ℵ 0 -space. Supplementing an old result of Corson we show that, for ǎ Cech-complete Lindelöf space X the following are equivalent: (a) X is Polish, (b) C c (X) is cosmic in the weak topology, (c) the weak * -dual of C c (X) is an ℵ 0 -space.

“…and say that U satisfies the condition (D) if U α = k∈N D k (α) for every α ∈ M. A similar condition naturally appears and is essentially used in [21].…”

Section: Description Of the Topology Of Cosmic And ℵ 0 -Spacesmentioning

confidence: 99%

“…Fact 1: (see [21]) AnyČech-complete Lindelöf space X has a compact resolution swallowing compact sets. Hence E = C c (X) has a G-base.…”

Section: The Following Corollary Provides a Nonmetrizable Counterpartmentioning

confidence: 99%

On topological properties of Fréchet locally convex spaces with the weak topology

Topology and its Applications

Kubzdela

et al. 2015

Self Cite

“…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].…”

Section: Introductionmentioning

confidence: 99%

Free locally convex spaces with a small base

2016

RACSAM

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.

“…and say that U satisfies the condition (D) if U α = k∈N D k (α) for every α ∈ M. A similar condition naturally appears and is essentially used in [15,17,19,18]. The above notion has been used to describe the topology of cosmic spaces, ℵ 0 -spaces and P 0spaces, respectively.…”

Section: Introductionmentioning

confidence: 99%

On topological spaces and topological groups with certain local countable networks

Topology and its Applications

2015

Self Cite

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.