Topological properties of function spaces C(X,2) over zero-dimensional metric spaces X

Gabriyelyan, Saak

doi:10.1016/j.topol.2016.05.021

Cited by 8 publications

(9 citation statements)

References 22 publications

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“…In [9] the first named author described all zero-dimensional metric spaces X for which the space C k (X, 2) is Ascoli, where 2 = {0, 1} is the doubleton.…”

Section: Theorem 13mentioning

confidence: 99%

The Ascoli property for function spaces

Gabriyelyan

Grebík

Ka̧kol

et al. 2016

Topology and its Applications

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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.

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“…In [9] the first named author described all zero-dimensional metric spaces X for which the space C k (X, 2) is Ascoli, where 2 = {0, 1} is the doubleton.…”

Section: Theorem 13mentioning

confidence: 99%

The Ascoli property for function spaces

Gabriyelyan

Grebík

Ka̧kol

et al. 2016

Topology and its Applications

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“…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

Gabriyelyan

2017

Topology and its Applications

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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 .

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“…The study of topological properties of spaces C p (X, Y ) and C k (X, Y ), when X = R, [0, 1] or the doubleton 2 = {0, 1}, is one of the main direction in General Topology. We refer the reader to the books [1,26,33] and the papers [5,16,11,18,19,20,28,29,30] and references therein.…”

Section: Sequentially Ascoli Function Spacesmentioning

confidence: 99%

“…The Ascoli property in topological spaces, topological groups, (free) locally convex spaces, function spaces etc. was intensively studied in [3,5,11,13,14,16,17,18,19]. Being motivated by the classical notion of c 0 -barrelled locally convex spaces, in [15] we defined a Tychonoff space X to be sequentially Ascoli if every convergent sequence in C k (X) is equicontinuous.…”

Section: Introductionmentioning

confidence: 99%

Ascoli and sequentially Ascoli spaces

Gabriyelyan¹

2020

Preprint

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A Tychonoff space X is called (sequentially) Ascoli if every compact subset (resp. convergent sequence) of C k (X) is evenly continuous, where C k (X) denotes the space of all real-valued continuous functions on X endowed with the compact-open topology. Various properties of (sequentially) Ascoli spaces are studied, and we give several characterizations of sequentially Ascoli spaces. Strengthening a result of Arhangel'skii we show that a hereditary Ascoli space is Fréchet-Urysohn. A locally compact abelian group G with the Bohr topology is sequentially Ascoli iff G is compact. If X is totally countably compact or near sequentially compact then it is a sequentially Ascoli space. The product of a locally compact space and an Ascoli space is Ascoli. If additionally X is a µ-space, then X is locally compact iff the product of X with any Ascoli space is an Ascoli space. Extending one of the main results of [18] and [16] we show that C p (X) is sequentially Ascoli iff X has the property (κ). We give a necessary condition on X for which the space C k (X) is sequentially Ascoli. For every metrizable abelian group Y , Y -Tychonoff space X, and nonzero countable ordinal α, the space B α (X, Y ) of Baire-α functions from X to Y is κ-Fréchet-Urysohn and hence Ascoli.

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Topological properties of function spaces C(X,2) over zero-dimensional metric spaces X

Cited by 8 publications

References 22 publications

The Ascoli property for function spaces

The Ascoli property for function spaces

On the Ascoli property for locally convex spaces

Ascoli and sequentially Ascoli spaces

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