Some weak covering properties and infinite games

Sakai, Masami

doi:10.2478/s11533-013-0343-4

Cited by 5 publications

(4 citation statements)

References 18 publications

(11 reference statements)

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“…We remarked that the assumption of countable closed pseudocharacter in Theorem 2.1 is satisfied in the case of a Hausdorff first-countable space. However the closed pseudocharacter of T 1 first-countable spaces can be arbitrarily large, and in fact Sakai showed in [12] an example of a first-countable T 1 space X of arbitrarily large cardinality such that player two has a winning strategy in G ω 1 (O X , O X ). Example 2.18.…”

Section: Variants Of Arhangel'skii's Theoremsmentioning

confidence: 99%

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Infinite games and cardinal properties of topological spaces

Bella¹,

Spadaro²

2012

Preprint

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Inspired by work of Scheepers and Tall, we use properties defined by topological games to provide bounds for the cardinality of topological spaces. We obtain a partial answer to an old question of Bell, Ginsburg and Woods regarding the cardinality of weakly Lindelöf first-countable regular spaces and answer a question recently asked by Babinkostova, Pansera and Scheepers. In the second part of the paper we study a game-theoretic version of cellularity, a special case of which has been introduced by Aurichi. We obtain a game-theoretic proof of Shapirovskii's bound for the number of regular open sets in an (almost) regular space and give a partial answer to a natural question about the productivity of a game strengthening of the countable chain condition that was introduced by Aurichi. As a final application of our results we prove that the Hajnal-Juhász bound for the cardinality of a first-countable ccc Hausdorff space is true for almost regular (non-Hausdorff) spaces.

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Section: Variants Of Arhangel'skii's Theoremsmentioning

confidence: 99%

“…Example 2.18. (Sakai, [12]) There is a T 1 almost regular first countable space of arbitrarily large cardinality X where player two has a winning strategy in G ω 1 (O X , O X ).…”

Section: Variants Of Arhangel'skii's Theoremsmentioning

confidence: 99%

Infinite games and cardinal properties of topological spaces

Bella¹,

Spadaro²

2012

Preprint

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“…e papers [3,4] carried out a systematic study of selection principles in topology and then research in this field expanded immensely and attracted many researchers (see survey papers [5][6][7] and references therein). Some types of selection principles (so-called weak selection principles) have been formulated by applying the interior and closure operators in the definition of a selection property (see [8][9][10][11][12][13][14][15][16][17][18][19][20][21]) and the other types have been explored by replacing sequences of open covers by sequences of covers by some generalized open sets (see [22][23][24]). In this paper, we apply the ideas from selection principles theory to soft topological spaces.…”

Section: Introductionmentioning

confidence: 99%

Nearly Soft Menger Spaces

Al-shami

Kočinac

2020

Journal of Mathematics

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In this paper, we define a weak type of soft Menger spaces, namely, nearly soft Menger spaces. We give their complete description using soft s-regular open covers and prove that they coincide with soft Menger spaces in the class of soft regular⋆ spaces. Also, we study the role of enriched and soft regular spaces in preserving nearly soft Mengerness between soft topological spaces and their parametric topological spaces. Finally, we establish some properties of nearly soft Menger spaces with respect to hereditary and topological properties and product spaces.

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“…In [14] we showed that there exists a topological space which is almost Menger and not Lindelöf and therefore, not Menger. Recently, Sakai in [19] gave an example of a topological space which is Lindelöf and not weakly Menger. So, in the previous diagram no other implication holds between given notions.…”

Section: Introduction and Notationmentioning

confidence: 99%

Menger-type covering properties of topological spaces

Kocev

2015

Filomat

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In this paper we find conditions under which the properties of Menger, almost Menger and weakly Menger are equivalent as well as the corresponding properties of Lindelöf-type. We give counterexamples that show the interrelations between those properties. The subject of our investigation is also the preservation of almost Menger and weakly Menger properties under subspaces and products. We also consider the weaker versions of Alster space and D-spaces.

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Some weak covering properties and infinite games

Cited by 5 publications

References 18 publications

Infinite games and cardinal properties of topological spaces

Infinite games and cardinal properties of topological spaces

Nearly Soft Menger Spaces

Menger-type covering properties of topological spaces

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