On sequentiality of spaces of continuous functions

Pytkeev, E. G.

doi:10.1070/rm1982v037n05abeh004037

Cited by 20 publications

(10 citation statements)

References 2 publications

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“…Gerlits [6], and independently Pytkeev [19], proved that if limits determine the closure in C(X), then indeed it suffices to take limits once. Theorem 1.1 (Gerlits, Pytkeev).…”

Section: Introduction and Basic Resultsmentioning

confidence: 99%

Pointwise convergence of partial functions: The Gerlits–Nagy Problem

Orenshtein

Tsaban

2013

Advances in Mathematics

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For a set X ⊆ R, let B(X) ⊆ R X denote the space of Borel real-valued functions on X, with the topology inherited from the Tychonoff product R X . Assume that for each countable A ⊆ B(X), each f in the closure of A is in the closure of A under pointwise limits of sequences of partial functions. We show that in this case, B(X) is countably Fréchet-Urysohn, that is, each point in the closure of a countable set is a limit of a sequence of elements of that set. This solves a problem of Arnold Miller. The continuous version of this problem is equivalent to a notorious open problem of Gerlits and Nagy. Answering a question of Salvador Hernańdez, we show that the same result holds for the space of all Baire class 1 functions on X.We conjecture that, in the general context, the answer to the continuous version of this problem is negative, but we identify a nontrivial context where the problem has a positive solution.The proofs establish new local-to-global correspondences, and use methods of infinitecombinatorial topology, including a new fusion result of Francis Jordan.

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“…Gerlits [6], and independently Pytkeev [19], proved that if limits determine the closure in C(X), then indeed it suffices to take limits once. Theorem 1.1 (Gerlits, Pytkeev).…”

Section: Introduction and Basic Resultsmentioning

confidence: 99%

Pointwise convergence of partial functions: The Gerlits–Nagy Problem

Orenshtein

Tsaban

2013

Advances in Mathematics

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“…In 1982 Pytkeev [20] and independently Gerlits [16] proved Theorem 6.1. (Pytkeev-Gerlits) For a space X, the following are equivalent:…”

Section: Modification Of Theorem P5mentioning

confidence: 98%

On generalization of theorems of Pestryakov

V.¹

2019

Preprint

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In 1987 A.V. Pestriakov proved a series of theorems for cardinal functions of the space B α (X) of all real-valued functions of Baire class α (α > 0), and he conjectured that most of these theorems are true for spaces containing all finite linear combinations of characteristic functions of zero-sets in X. In this paper we investigate for which theorems of Pestriakov generalizations are valid. Also we prove some additional propositions for function spaces applying the theory of selection principles.

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“…We prove the following result which also shows that the Fréchet-Urysohn property, sequentiality and k-space property differ on spaces of the form C k (X, 2). While these properties coincide for C k (X) by Pytkeev's theorem [25]: for a Tychonoff space X, the space C k (X) is a k-space if and only if it is Fréchet-Urysohn. Theorem 1.4.…”

Section: Introductionmentioning

confidence: 99%

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

Gabriyelyan

2016

Topology and its Applications

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Let X be a zero-dimensional metric space and X ′ its derived set. We prove the following assertions: (1) the space C k (X, 2) is an Ascoli space iff C k (X, 2) is k R -space iff either X is locally compact or X is not locally compact but X ′ is compact, (2) C k (X, 2) is a k-space iff either X is a topological sum of a Polish locally compact space and a discrete space or X is not locally compact but X ′ is compact, (3) C k (X, 2) is a sequential space iff X is a Polish space and either X is locally compact or X is not locally compact but X ′ is compact, (4) C k (X, 2) is a Fréchet-Urysohn space iff C k (X, 2) is a Polish space iff X is a Polish locally compact space, (5) C k (X, 2) is normal iff X ′ is separable, (6) C k (X, 2) has countable tightness iff X is separable. In cases (1)-(3) we obtain also a topological and algebraical structure of C k (X, 2).

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On sequentiality of spaces of continuous functions

Cited by 20 publications

References 2 publications

Pointwise convergence of partial functions: The Gerlits–Nagy Problem

Pointwise convergence of partial functions: The Gerlits–Nagy Problem

On generalization of theorems of Pestryakov

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

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