Relating games of Menger, countable fan tightness, and selective separability

Clontz, Steven

doi:10.1016/j.topol.2019.02.062

Cited by 3 publications

(8 citation statements)

References 7 publications

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“…The equivalence (2) ⇔ (3) is similar to the following open question of Gruenhage, first shown to be true when X is countable by Barman and Dow in [5,Theorem 2.11]; see [10,Lemma 37] for a general sufficient condition which guarantees that a winning strategy may be improved to a Markov winning strategy.…”

Section: Has a Countable Dense Subset D Where Twomentioning

confidence: 70%

“…Proof. In [10], Φ = D was an additional assumption, but was never required in the proofs, since S * (Φ, D) implies separability for any non-empty Φ.…”

Section: Is Separable and Satisfies Smentioning

confidence: 99%

“…In the papers [2,3,6,16,[22][23][24][25][26]32] various selection principles for a Tychonoff space X were related to the selection principles for C p (X). Likewise, in [10,17,26,30,31] various selection games for X and C p (X) and a bitopological space (C(X), τ k , τ p ) were related.…”

Section: Theorem 43 ([20]mentioning

confidence: 99%

“…Second-countability allows us to apply Corollary 4.5 to show (1-3) are mutually equivalent, as are (4)(5)(6). By [10, Corollary 39], ( 3) is equivalent to (6), and by [10,Lemma 24], (6) equivalent to (7).…”

Section: Theorem 43 ([20]mentioning

confidence: 99%

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On Ramsey properties, function spaces, and topological games

Clontz

Osipov

2020

Filomat

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An open question of Gruenhage asks if all strategically selectively separable spaces are Markov selectively separable, a game-theoretic statement known to hold for countable spaces. As a corollary of a result by Berner and Juh?sz, we note that the ?strong? version of this statement, where the second player is restricted to selecting single points rather than finite subsets, holds for all T3 spaces without isolated points. Continuing this investigation, we also consider games related to selective sequential separability, and demonstrate results analogous to those for selective separability. In particular, strong selective sequential separability in the presence of the Ramsey property may be reduced to a weaker condition on a countable sequentially dense subset. Additionally, ?- and ?-covering properties on X are shown to be equivalent to corresponding sequential properties on Cp(X). A strengthening of the Ramsey property is also introduced, which is still equivalent to ?2 and ?4 in the context of Cp(X).

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Section: Has a Countable Dense Subset D Where Twomentioning

confidence: 70%

“…Proof. In [10], Φ = D was an additional assumption, but was never required in the proofs, since S * (Φ, D) implies separability for any non-empty Φ.…”

Section: Is Separable and Satisfies Smentioning

confidence: 99%

Section: Theorem 43 ([20]mentioning

confidence: 99%

Section: Theorem 43 ([20]mentioning

confidence: 99%

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On Ramsey properties, function spaces, and topological games

Clontz

Osipov

2020

Filomat

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“…Classic works by Telgarksy and Galvin show that the point open game is dual to the Rothberger game [4]. Clontz, in work prior to this, established the equivalence of the existence of winning strategies for the Rothberger game and variants of the Rothberger game on X to the existence of winning strategies in games related to countable fan tightness for C p (X) [5]. Clontz did this both for strategies of perfect information and for limited information strategies.…”

mentioning

confidence: 99%

Limited information strategies and discrete selectivity

Clontz

Holshouser

2019

Topology and its Applications

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We relate the property of discrete selectivity and its corresponding game, both recently introduced by V.V.Tkachuck, to a variety of selection principles and point picking games. In particular we show that player II can win the discrete selection game on C p (X) if and only if player II can win a variant of the point open game on X. We also show that the existence of limited information strategies in the discrete selection game on C p (X) for either player are equivalent to other well-known topological properties.In the course of studying the strong domination of function spaces by second countable spaces and countable spaces, G. Sanchez and Tkachuk isolated the topological property of discrete selectivity[1] [2]. A space is discretely selective if for every sequence {U n : n ∈ ω} of non-empty open subsets of the space, there are points x n ∈ U n so that {x n : n ∈ ω} is closed discrete. In subsequent work, Tkachuk showed that for T 3.5 -spaces, C p (X) is discretely selective if and only if X is uncountable.Discrete selectivity naturally generates a game, in which player I plays open sets, player II responds with points from those open sets, and player II wins if the points form a closed discrete set. Tkachuk explored what happens when player I has a winning strategy for this game, showing that the existence of a winning strategy for player I in this game on C p (X) is equivalent to player I having a winning strategy for Gruenhage's W -game on C p (X, 0) and is also equivalent to player I having a winning strategy for the pointopen game on X [3]. Tkachuk also showed that if player II has a winning strategy in the point-open game on X, then player II has a winning strategy in the discrete selection game on C p (X). Tkachuk hypothesized that the implication partially reverses for player II (considering ω-covers), and posed this problem as an open question. All of the strategies Tkachuk worked with were perfect information strategies.By considering limited information strategies and other topological games, we were able to answer

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On Ramsey properties, function spaces, and topological games

Clontz¹,

Osipov²

2019

Preprint

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An open question of Gruenhage asks if all strategically selectively separable spaces are Markov selectively separable, a game-theoretic statement known to hold for countable spaces. As a corollary of a result by Berner and Juhász, we note that the "strong" version of this statement, where the second player is restricted to selecting single points in the rather than finite subsets, holds for all T 3 spaces without isolated points. Continuing this investigation, we also consider games related to selective sequential separability, and demonstrate results analogous to those for selective separability. In particular, strong selective sequential separability in the presence of the Ramsey property may be reduced to a weaker condition on a countable sequentially dense subset. Additionally, γ-and ω-covering properties on X are shown to be equivalent to corresponding sequential properties on C p (X). A strengthening of the Ramsey property is also introduced, which is still equivalent to α 2 and α 4 in the context of C p (X).

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Relating games of Menger, countable fan tightness, and selective separability

Cited by 3 publications

References 7 publications

On Ramsey properties, function spaces, and topological games

On Ramsey properties, function spaces, and topological games

Limited information strategies and discrete selectivity

On Ramsey properties, function spaces, and topological games

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