Topological Games Research Articles

Star γ Covers and Their Applications to Selection Principles

Abstract By imposing the star operator on γ covers, we establish the notion of st-γ covers. The interrelationships among many recent variants of γ covers have been identified, and various counterexamples have been shown to distinguish them. An investigation has been conducted on the basic topological attributes of these covers. We have also examined the implications between various selection principles and topological games in the context of the class of st-γ coverings.

Certain Observations on Tightness and Topological Games in Bornology

Certain Observations on Tightness and Topological Games in Bornology

On some topological games involving networks

In these notes we introduce and investigate two new games called R-nw-selective game and the M-nw-selective game. These games naturally arise from the corresponding selection principles involving networks introduced in [5].

Consonant spaces of countable type and the Menger property

We investigate the connection between consonance and the Menger property. It has been shown that consonant metric spaces are Menger at infinity. We improve this result by showing that completely regular consonant spaces of countable type are Menger at infinity. This result will follow from a number of results about topological games in spaces that are more general than completely regular spaces.

Generalization of the Grothendieck's theorem

In this paper, we obtain a generalization of Grothendieck's theorem for the space of continuous mappings Cλ,μ(X,Y) where Y is a complete uniform space with the uniformity μ endowed with the topology of uniform convergence on the family λ of subsets of X. A new topological game is defined - the Asanov-Velichko game, which makes it possible to single out a class of topological spaces of the Grothendieck type.The developed technique is used to generalize the Grothendieck's theorem for the space of continuous mappings endowed with the set-open topology.

Abstract colorings, games and ultrafilters

The main result provides a common generalization for Ramsey-type theorems concerning finite colorings of edge sets of complete graphs with vertices in infinite semigroups. We capture the essence of theorems proved in different fields: for natural numbers due to Milliken–Tylor, Deuber–Hindman, Bergelson–Hindman, for combinatorial covering properties due to Scheepers and Tsaban, and local properties in function spaces due to Scheepers. To this end, we use idempotent ultrafilters in the Čech–Stone compactifications of discrete infinite semigroups and topological games. The research is motivated by the recent breakthrough work of Tsaban about colorings and the Menger covering property.

On Conway’s Brussels Sprouts

John Horton Conway stood out from many famous mathematicians for his love of games and puzzles. Among others, he is known for inventing the two-player topological games called Sprouts and Brussels Sprouts. These games start with n spots (n crosses resp.), have simple rules, last for finitely many moves, and the player who makes the last move wins. In the misère versions, the player who makes the last move loses. In this paper, we make Brussels Sprouts colored, preserving the aesthetic interest and balance of the game. In contrast to the original Sprouts, Colored Brussels Sprouts allows mathematical analysis without computer programming and has winning strategies for a large family of the number of spots.

Classes of Baire spaces defined by topological games

Classes of Baire spaces defined by topological games

Fuzzy Mathematical Modelling of Stochastic Games

In this paper, we present about the fuzzy mathematical modelling of stochastic games. By introducing the concept of topological game over an ideal of Hausdorff space, a game over some special product space is played. Fuzzy set theory has been applied to fuzzify some of the results obtained.

Fuzzy Mathematical Modelling and Lexicographical Solution Concept in Fuzzy Games

In this paper, we present about the fuzzy mathematical modelling and lexicographical solution concept in fuzzy games. By introducing the concept of topological game over an ideal of Hausdorff space, a game over some special product space is played. Fuzzy set theory has been applied to fuzzify some of the results obtained.

A note on the Freese–Nation property and topological games

A note on the Freese–Nation property and topological games

On irresolute multifunctions and related topological games

<abstract><p>In this paper, we introduce and study $ \alpha $-irresolute multifunctions, and some of their properties are studied. The properties of $ \alpha $-compactness and $ \alpha $-normality under upper $ \alpha $-irresolute multifunctions are topological properties. Also, we prove that the composition of two upper and lower $ \alpha $-irresolute multifunctions is $ \alpha $-irresolute. We apply the results of $ \alpha $-irresolute multifunctions to topological games. Upper and lower topological games are introduced. The set of places for player ONE in upper topological games may guarantee a gain is semi-closed. Finally, some optimal strategies for topological games are defined and studied.</p></abstract>

Some observations on the mildly Menger property and topological games

In this paper, we defined two new games - the mildly Menger game and the compact-clopen game. In a zero-dimensional space, the Menger game is equivalent to the mildly Menger game and the compact-open game is equivalent to the compact-clopen game. An example is given for a space on which the mildly Menger game is undetermined. Also we introduced a new game namely K-quasi-component-clopen game and proved that this game is equivalent to the compact-clopen game. Then we proved that if a topological space is a union of countably many quasi-components of compact sets, then TWO has a winning strategy in the mildly Menger game.

Joint continuity of separately continuous mappings

In this paper, we will introduce a topological game to show that every separately continuous function f:X×Y→Z, where X is a β-unfavorable space for our game and Z is a regular space, is strongly quasi-continuous at each point of X×{y} provided that y is a q-point. We also show that the set of points of subcontinuity of quasi-continuous mappings from a Baire space X into a regular game determined space Y contains a dense Gδ subset of X. Our results generalize some theorems in Bouziad (1996) [6] and Kenderov et al. (2001) [18].

Pointless proofs of the Menger and Rothberger games

We present proofs of Hurewicz's and Pawlikowski's Theorems concerning topological games, but in the context of lattices, in which one cannot argue with points.

The Baire property on the hyperspace of nontrivial convergent sequences

In this paper, we shall consider the hyperspace of all nontrivial convergent sequences Sc(X) of a Fréchet-Urysohn nondiscrete space X, which is equipped with the Vietoris topology. We study the spaces X for which Sc(X) is Baire: this kind of spaces have a dense subset of isolated points (see [12]). We characterize the spaces X for which Sc(X) is Baire. This characterization uses a topological game inspired by the Banach-Mazur game. As a consequence, we obtain that if X is completely metrizable and has a dense subset of isolated points, then Sc(X) is Baire. Our last main result shows that Sc(X) is pseudocompact iff X is homeomorphic to the one-point compactification of a discrete space of uncountable size. This last assertion provides a characterization of the one-point compactification of a discrete space of uncountable size.

Topological games of bounded selections

We present a new variation of the classical selection principles Sk(A,B) (k∈N) and Sfin(A,B) that formally lies between these two properties. As in the case of the classical selection principles, we also obtain a new variation of topological game and discuss how new topological properties may emerge in the specific cases of covering and tightness.

Games relating to weak covering properties in bitopological spaces

We study topological games related to weak forms of the Menger property in bitopological spaces. In particular we investigate almost Menger game and its connections to games which are associated with the covering properties consisting of covers containing $G_\delta$ subsets.

Points of Continuity of Quasi-Continuous Functions

Abstract We study points of joint continuity of multi-variable separately quasi-continuous functions which are continuous with respect to one variable. We will show that the assumption of complete regularity in the paper of V. Maslyuchenko et al in ”Tatra Mt. Math. Pub. 68, (2017), 47–58” can be removed in some situations. Moreover, we use a topological game argument to prove that the set of points of continuity of functions with closed graph into ωΔ-spaces is a Gδ subset of its domain provided that its domain is a W-space.

Meager sets, games and singular cardinals

We show that a statement concerning the existence of winning strategies of limited memory in an infinite two-person topological game is equivalent to a weak version of the Singular Cardinals Hypothesis.