The Reznichenko property and the Pytkeev property in hyperspaces

Koɩinac, Lj. D. R.

doi:10.1007/s10474-005-0192-0

Cited by 20 publications

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“…The strong Pytkeev property is usually studied combining the other spaces, such as topological groups, topological vector spaces, etc., see [11,20,21,28,32]. In this paper, we mainly research the strong Pytkeev property in topological gyrogroups.…”

Section: Introductionmentioning

confidence: 99%

The strong Pytkeev property and strong countable completeness in (strongly) topological gyrogroups

Meng¹,

Zhang²,

Xu³

2021

Preprint

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A topological gyrogroup is a gyrogroup endowed with a topology such that the binary operation is jointly continuous and the inverse mapping is also continuous. In this paper, it is proved that if G is a sequential topological gyrogroup with an ω ω -base, then G has the strong Pytkeev property. Moreover, some equivalent conditions about ω ω -base and strong Pytkeev property are given in Baire topological gyrogroups. Finally, it is shown that if G is a strongly countably complete strongly topological gyrogroup, then G contains a closed, countably compact, admissible subgyrogroup P such that the quotient space G/P is metrizable and the canonical homomorphism π : G → G/P is closed.

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Section: Introductionmentioning

confidence: 99%

The strong Pytkeev property and strong countable completeness in (strongly) topological gyrogroups

Meng¹,

Zhang²,

Xu³

2021

Preprint

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“…This paper is devoted to a systematic study of the strong Pytkeev property in topological spaces. The strong Pytkeev property was introduced by Tsaban and Zdomskyy in [33] and is a hereditary version of the Pytkeev property introduced by Pytkeev in [25] and studied in [6], [8], [20], [22], [24], [26], [27], [32]. In Section 1 of this paper we discuss the interplay between the strong Pytkeev property and other known local properties of topological spaces, in Sections 2 detect function spaces with the strong Pytkeev property, in Section 3 we apply the results of Section 2 to establish the stability the class of topological spaces with the strong Pytkeev under countable Tychonoff products and small box-products.…”

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The strong Pytkeev property in topological spaces

Банах

Leiderman

2017

Topology and its Applications

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Modifying the known definition of a Pytkeev network, we introduce a notion of Pytkeev * network and prove that a topological space has a countable Pytkeev network if and only if X is countably tight and has a countable Pykeev * network at x. In the paper we establish some stability properties of the class of topological spaces with the strong Pytkeev * -property.1991 Mathematics Subject Classification. 54D70, 54E18. Key words and phrases. Pytkeev network, the strong Pytkeev property, local ω ω -base, k-network.Corollary 2.6 and Lemmas 3.3, 3.5 imply the following characterizations.Corollary 3.7. A topological space X has a countable Pytkeev network at a point x ∈ X if and only if X has a countable Pytkeev * network at x and X is countably tight at x. Corollary 3.8. A topological space X has a countable Pytkeev network if and only if X has a countable Pytkeev * network.Proof. The "only if" part follows from Lemma 3.5. To prove the "if" part, assume that X has a countable Pytkeev * network. Then X has a countable network and hence is hereditarily separable and countably tight. By Corollary 3.4, the space X has a countable Pytkeev network.Finally we present an example of a Pytkeev * -network which is not a Pytkeev network.A point x of a topological space X is a weak P -point if x / ∈Ā for any countable set A ⊂ X \ {x}. By [19, 4.3.4] the remainder βω \ ω of the Stone-Čech compactification βω of ω contains a non-isolated weak P -point.Example 3.9. If X is a non-isolated weak P -point in a topological space, then the family N = {x} is a Pytkeev * network at x, which is not a Pytkeev network at x. The strong Pytkeev * property versus the strong Pytkeev propertyIn this section we recall the definition of the strong Pytkeev property, introduce the strong Pytkeev * property and study the interplay between these two notions.Definition 4.1. A topological space X is defined to have• the strong Pytkeev property at a point x ∈ X if X has a countable Pytkeev network at x ∈ X;• the strong Pytkeev property if X has the strong Pytkeev property at each point x ∈ X. The strong Pytkeev property was introduced in [22] and thoroughly studied in [20], [12], [6]. Replacing in Definition 4.2 Pytkeev networks by Pytkeev * networks, we obtain the definition of the strong Pytkeev * property. Definition 4.2. A topological space X is defined to have• the strong Pytkeev * property at a point x ∈ X if X has a countable Pytkeev * network at x ∈ X;• the strong Pytkeev * property if X has the strong Pytkeev * property at each point x ∈ X.Corollary 3.7 implies that strong Pytkeev property decomposes into the combination of the countable tightness and the strong Pytkeev * property.

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