Point-open Game Research Articles

Some observations on a clopen version of the Rothberger property

In this paper, we prove that a clopen version $S_1(\mathcal{C}_\mathcal{O}, \mathcal{C}_\mathcal{O})$ of the Rothberger property and Borel strong measure zeroness are independent. For a zero-dimensional metric space $(X,d)$, $X$ satisfies $S_1(\mathcal{C}_\mathcal{O}, \mathcal{C}_\mathcal{O})$ if, and only if, $X$ has Borel strong measure zero with respect to each metric which has the same topology as $d$ has. In a zero-dimensional space, the game $G_1(\mathcal{O}, \mathcal{O})$ is equivalent to the game $G_1(\mathcal{C}_\mathcal{O}, \mathcal{C}_\mathcal{O})$ and the point-open game is equivalent to the point-clopen game. Using reflections, we obtain that the game $G_1(\mathcal{C}_\mathcal{O}, \mathcal{C}_\mathcal{O})$ and the point-clopen game are strategically and Markov dual. An example is given for a space on which the game $G_1(\mathcal{C}_\mathcal{O}, \mathcal{C}_\mathcal{O})$ is undetermined.

Selection games on continuous functions

In this paper we study the selection principle of closed discrete selection, first researched by Tkachuk in [12] and strengthened by Clontz, Holshouser in [3], in set-open topologies on the space of continuous real-valued functions. Adapting the techniques involving point-picking games on X and Cp(X), the current authors showed similar equivalences in [1] involving the compact subsets of X and Ck(X). By pursuing a bitopological setting, we have touched upon a unifying framework which involves three basic techniques: general game duality via reflections (Clontz), general game equivalence via topological connections, and strengthening of strategies (Pawlikowski and Tkachuk). Moreover, we develop a framework which identifies topological notions to match with generalized versions of the point-open game.

Dual selection games

Often, a given selection game studied in the literature has a known dual game. In dual games, a winning strategy for a player in either game may be used to create a winning strategy for the opponent in the dual. For example, the Rothberger selection game involving open covers is dual to the point-open game. This extends to a general theorem: if {range(f):f∈C(R)} is coinitial in A with respect to ⊆, where C(R)={f∈(⋃R)R:R∈R⇒f(R)∈R} collects the choice functions on the set R, then G1(A,B) and G1(R,¬B) are dual selection games.

Limited information strategies and discrete selectivity

We relate the property of discrete selectivity and its corresponding game, both recently introduced by V.V. Tkachuk, to a variety of selection principles and point-picking games. In particular we show that player II can win the discrete selectivity game on Cp(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 selectivity game on Cp(X) for either player are equivalent to other well-known topological properties.

On generalizations of topological games

We establish three games as generalizations of the Banach-Mazur game, the Choquet game and the point-open game. We find a characterization of compact scattered spaces by using one of the generalized games and show a relation between the other generalized games and domain representability.

Some applications of the point-open subbase game

Given a subbase $\mathcal S$ of a space $X$, the game $PO(\mathcal S,X)$ is defined for two players $P$ and $O$ who respectively pick, at the $n$-th move, a point $x_n\in X$ and a set $U_n\in \mathcal S$ such that $x_n\in U_n$. The game stops after the moves $\{x_n,U_n: n\in\o\}$ have been made and the player $P$ wins if $\bigcup_{n\in\o}U_n=X$; otherwise $O$ is the winner. Since $PO(\mathcal S,X)$ is an evident modification of the well-known point-open game $PO(X)$, the primary line of research is to describe the relationship between $PO(X)$ and $PO(\mathcal S,X)$ for a given subbase $\mathcal S$. It turns out that, for any subbase $\mathcal S$, the player $P$ has a winning strategy in $PO(\mathcal S,X)$ if and only if he has one in $PO(X)$. However, these games are not equivalent for the player $O$: there exists even a discrete space $X$ with a subbase $\mathcal S$ such that neither $P$ nor $O$ has a winning strategy in the game $PO(\mathcal S,X)$. Given a compact space $X$, we show that the games $PO(\mathcal S,X)$ and $PO(X)$ are equivalent for any subbase $\mathcal S$ of the space $X$.

Rothberger bounded groups and Ramsey theory

We show that: (1) Rothberger bounded subgroups of σ-compact groups are characterized by Ramseyan partition relations (Corollary 4). (2) For each uncountable cardinal κ there is a T 0 topological group of cardinality κ such that ONE has a winning strategy in the point-open game on the group and the group is not a closed subspace of any σ-compact space (Theorem 8). (3) For each uncountable cardinal κ there is a T 0 topological group of cardinality κ such that ONE has a winning strategy in the point-open game on the group and the group is σ-compact (Corollary 17).

One more variation of the point-open game

A topological game Dense G��-sets (also denoted by DG) is introduced as follows: for any n 2 ! at the n-th move the player I takes a point xn 2 X and II responds by taking a G�-set Qn in the space X such that xn 2 Qn. The play stops after ! moves and I wins if the set S {Qn : n 2 !} is dense in X. Otherwise the player II is declared to be the winner. We study classes of spaces on which the player I has a winning strategy. It is evident that the I-favorable spaces constitute a generalization of the class of separable spaces. We show that there exists a neutral space for the game DG and prove, among other things, that Lindelof scattered spaces and dyadic spaces are I-favorable. We characterize I- favorability for the game DG in the spaces Cp(X); one of the applications is that, for a Lindelof �-space X, the space Cp(X) is I-favorable for DG if and only if X is !-monolithic.

Every Lusin set is undetermined in the point-open game

We show that some classes of small sets are topological versions of some combinatorial properties. We also give a characterization of spaces for which White has a winning strategy in the point-open game. We show that every Lusin set is undetermined, which

Undetermined sets of point-open games

We show that a set of reals is undetermined in Galvin's point-open game iff it is uncountable and has property C", which answers a question of Gruenhage.

Possible point-open types of subsets of the reals

If X is a topological space and α is an ordinal, then the point-open game of length α on X, abbreviated G α( X), is the two person game of length α in which, on the βth move (β<α), the first player (the “point picker”) picks a point of X and the second player (the “open set picker”) picks an open subset of X covering the point just played. The point picker wins iff the open sets thus picked cover X. The point-open type of X, abbreviated pot( X), is defined to be the smallest ordinal α such that the point picker has a winning strategy in G α( X). This ordinal clearly exists and is no more than the cardinality of X. The main result of this paper is that if we assume the Continuum Hypothesis, then for every limit ordinal α<ω 1, there is a subset X of the real numbers such that pot ( X) = α. This solves a problem due to Peg Daniels and Gary Gruenhage.

The point-open type of subsets of the reals

The point-open type, pot( X), of a space X is defined to be the least ordinal α such that the point-picker has a winning strategy in the point-open game of lenght α. We establish the basic properties of pot( X), show that pot( X)⩽ω 1 for each hereditarily Lindelöf space X, characterize those subsets X of the real line R for which pot( X)<ω 2, give assuming the continuum hypothesis an example for each n⩽ω of a set X⊂ R with pot( X)=ω· n, and discuss the question of what values of pot( X) are possible for X⊂ R .

A Singular Space Related to the Point-Open Game

The authors construct a completely regular space ${X_0}$ such that Player I has a winning strategy in the point-open game $G\left ( {{X_0}} \right )$, but ${X_0}$ has no $\sigma$-closure-preserving cover by ${\mathbf {C}}$-scattered closed sets.

A singular space related to the point-open game

The authors construct a completely regular space X 0 {X_0} such that Player I has a winning strategy in the point-open game G ( X 0 ) G\left ( {{X_0}} \right ) , but X 0 {X_0} has no σ \sigma -closure-preserving cover by C {\mathbf {C}} -scattered closed sets.