On the $$C_k$$ C k -stable closure of the class of (separable) metrizable spaces

Descriptive Topology and Functional Analysis II

2019

For a Tychonoff space X, denote by P the family of topological properties P of being a convergent sequence or being a compact, sequentially compact, countably compact, pseudocompact and functionally bounded subset of X, respectively. A maximally almost periodic (M AP ) group G respects P if P(G) = P(G + ), where G + is the group G endowed with the Bohr topology. We study relations between different respecting properties from P and show that the respecting convergent sequences (=the Schur property) is the weakest one among the properties of P. We characterize respecting properties from P in wide classes of M AP topological groups including the class of metrizable M AP abelian groups. Every real locally convex space (lcs) is a quotient space of an lcs with the Schur property, and every locally quasi-convex (lqc) abelian group is a quotient group of an lqc abelian group with the Schur property. It is shown that a reflexive group G has the Schur property or respects compactness iff its dual group G ∧ is c0-barrelled or g-barrelled, respectively. We prove that an lqc abelian kω-group respects all properties P ∈ P. As an application of the obtained results we show that (1) the space C k (X) is a reflexive group for every separable metrizable space X, and (2) a reflexive abelian group of finite exponent is a Mackey group. P 0 := {S, C, SC, CC, PC} and P := P 0 ∪ {FB}.Let G be a maximally almost periodic (M AP ) topological group G (for all relevant definitions, see Section 2). We denote by G + the group G endowed with the Bohr topology. Following [57], a M AP group G respects a topological property P if P(G) = P(G + ).The famous Glicksberg theorem [36] states that every locally compact abelian (LCA) group respects compactness. If a M AP group G respects compactness we shall say also that G has the Glicksberg property. Trigos-Arrieta [60,61] proved that countable compactness, pseudocompactness and functional boundedness are respected by LCA groups. Banaszczyk and Martín-Peinador [9] generalized these results to all nuclear groups. Nuclear groups were introduced and thoroughly studied by Banaszczyk in [8]. The concept of Schwartz topological abelian groups is appeared in [4]. This notion generalizes the well-known notion of a Schwartz locally convex space. All nuclear groups are Schwartz groups [4]. Außenhofer [2] proved that every locally quasi-convex Schwartz group respects compactness. For a general and simple approach to the theory of properties respected by M AP topological groups see [24].Let (E, τ ) be a locally convex space (lcs for short), E ′ the dual space of E and let τ w = σ(E, E ′ ) be the weak topology on E. Set E w := (E, τ w ). An lcs E is said to have the Schur property if E and E w have the same convergent sequences, i.e., S(E) = S(E w ). Considering E as an additive 2000 Mathematics Subject Classification. Primary 22A05, 46A03; Secondary 54H11.

Section: Real Locally Convex Spaces and Respecting Propertiesmentioning

confidence: 99%

Maximally Almost Periodic Groups and Respecting Properties

Descriptive Topology and Functional Analysis II

2019

“…McCoy proved in [25] that for a first countable paracompact X the space C k (X) is a k-space if and only if X is hemicompact, so C k (X) is metrizable. Being motivated by the classic Ascoli theorem we introduced in [4] a new class of topological spaces, namely, the class of Ascoli spaces. A Tychonoff space X is Ascoli if every compact subset of C k (X) is evenly continuous.…”

Section: Introductionmentioning

confidence: 99%

“…and none of these implications is reversible. The Ascoli property for function spaces has been studied recently in [3,4,10,12,13,16]. Let us mention the following Theorem 1.1.…”

Section: Introductionmentioning

confidence: 99%

On the Ascoli property for locally convex spaces

Topology and its Applications

2017

Self Cite

We characterize Ascoli spaces by showing that a Tychonoff space X is Ascoli iff the canonical map from the free locally convex space L(X) over X into C k C k (X) is an embedding of locally convex spaces. We prove that an uncountable direct sum of non-trivial locally convex spaces is not Ascoli. If a c0-barrelled space X is weakly Ascoli, then X is linearly isomorphic to a dense subspace of R Γ for some Γ. Consequently, a Fréchet space E is weakly Ascoli iff E = R N for some N ≤ ω. If X is a µ-space and a k-space (for example, metrizable), then C k (X) is weakly Ascoli iff X is discrete. We prove that the weak* dual space of a Banach space E is Ascoli iff E is finite-dimensional.2000 Mathematics Subject Classification. Primary 46A03; Secondary 54A25, 54D50. Key words and phrases. the Ascoli property, free locally convex space, direct sum, inductive limit, compactly barrelled space .

“…Following [3], a space X is called an Ascoli space if each compact subset K of C k (X) is evenly continuous, that is, the map X × K ∋ (x, f ) → f (x) ∈ R is continuous. Equivalently, X is Ascoli if the natural evaluation map X ֒→ C k (C k (X)) is an embedding, see [3].…”

Section: Introductionmentioning

confidence: 99%

“…Equivalently, X is Ascoli if the natural evaluation map X ֒→ C k (C k (X)) is an embedding, see [3]. Recall that a space X is called a k R -space if a real-valued function f on X is continuous if and only if its restriction f | K to any compact subset K of X is continuous.…”

Section: Introductionmentioning

confidence: 99%

The Ascoli property for function spaces

Topology and its Applications

Grebík

Ka̧kol

et al. 2016

Abstract. The paper deals with Ascoli spaces Cp(X) and C k (X) over Tychonoff spaces X. The class of Ascoli spaces X, i.e. spaces X for which any compact subset K of C k (X) is evenly continuous, essentially includes the class of k R -spaces. First we prove that if Cp(X) is Ascoli, then it is κ-Fréchet-Urysohn. If X is cosmic, then Cp(X) is Ascoli iff it is κ-Fréchet-Urysohn. This leads to the following extension of a result of Morishita: If for aČech-complete space X the space Cp(X) is Ascoli, then X is scattered. If X is scattered and stratifiable, then Cp(X) is an Ascoli space. Consequently: (a) If X is a complete metrizable space, then Cp(X) is Ascoli iff X is scattered. (b) If X is aČech-complete Lindelöf space, then Cp(X) is Ascoli iff X is scattered iff Cp(X) is Fréchet-Urysohn. Moreover, we prove that for a paracompact space X of point-countable type the following conditions are equivalent:is an Ascoli space. The Asoli spaces C k (X, I) are also studied.