Some properties of C(X), I

Gerlits, J.; Nagy, Zs.

doi:10.1016/0166-8641(82)90065-7

Cited by 258 publications

(277 citation statements)

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“…Therefore C p (X) is an Ascoli space by Theorem 2.3 and Corollary 1.2. However, it follows from the results of Gerlits and Nagy [12] that no unbounded subset of ω ω has property (γ), and hence C p (X) is not Fréchet-Urysohn. So C p (X) is not a k-space by the Pytkeev-Gerlitz-Nagy theorem.…”

Section: Theorem 13mentioning

confidence: 99%

See 1 more Smart Citation

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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Section: Theorem 13mentioning

confidence: 99%

“…The famous Pytkeev-Gerlitz-Nagy theorem, see [1,Theorem II.3.7], states that C p (X) is a k-space if and only if C p (X) is Fréchet-Urysohn if and only if X has the covering property (γ) introduced in [12]. Below we give an example of a separable metrizable space X for which C p (X) is Ascoli but is not a k-space.…”

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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“…It is known that under the Martin's axiom there are γ-sets of cardinality continuum [54]. On the other hand, every γ-set has a strong measure zero [56], so it is consistent with ZFC that every γ-set is countable. Moreover, CPA game cube implies that every γ-set has cardinality at most ω 1 < c, since every strong measure zero is universally null and so, by Theorem 1.1.4, under CPA cube every γ-set has cardinality at most ω 1 .…”

Section: Uncountable γ-Sets and Strongly Meager Setsmentioning

confidence: 64%

“…γ-sets were introduced by Gerlits and Nagy [56]. They were studied by Galvin and Miller [54], Rec law [100], Bartoszyński, Rec law [5], and others.…”

Section: Uncountable γ-Sets and Strongly Meager Setsmentioning

confidence: 99%

The Covering Property Axiom, CPA

Ciesielski

Pawlikowski

2004

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“…A and B will be collections of the following open covers of a space X: O: the collection of all open covers of X; Ω: the collection of ω-covers of X. An open cover U of X is an ω-cover [5] if X does not belong to U and every finite subset of X is contained in an element of U; Γ: the collection of γ-covers of X. An open cover U of X is a γ-cover [5] if it is infinite and each x ∈ X belongs to all but finitely many elements of U. O gp : the collection of groupable open covers.…”

Section: Introductionmentioning

confidence: 99%