Related papers: Topological games and Alster spaces
Let X be a Tychonoff space and MC(X) be the space of convex minimal usco maps with values in R, the space of real numbers. Such set-valued maps are important in the study of subdifferentials of convex functions. Using the strong Choquet…
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.
Menger's conjecture that Menger spaces are /sigma-compact is false; it is true for analytic subspaces of Polish spaces and undecidable for more complex definable subspaces of Polish spaces. For non-metrizable spaces, analytic Menger spaces…
XOR games are a simple computational model with connections to many areas of complexity theory. Perhaps the earliest use of XOR games was in the study of quantum correlations. XOR games also have an interesting connection to Grothendieck's…
The radius-$r$ splitter game is played on a graph $G$ between two players: Splitter and Connector. In each round, Connector selects a vertex $v$, and the current game arena is restricted to the radius-$r$ neighborhood of $v$. Then Splitter…
Let $T$ be a bounded linear operator on a Hilbert space. Then the Aluthge transform $\Delta T$ and the sequence $(\Delta^nT)$ of Aluthge iterates of $T$ are defined by \begin{align*} \Delta…
Colonel Blotto games with discrete strategy spaces effectively illustrate the intricate nature of multidimensional strategic reasoning. This paper studies the equilibrium set of such games where, in line with prior experimental work, the…
Every topological group $G$ has some natural compactifications which can be a useful tool of studying $G$. We discuss the following constructions: (1) the greatest ambit $S(G)$ is the compactification corresponding to the algebra of all…
Let X be a h-homogeneous zero-dimensional compact Hausdorff space, i.e. X is a Stone dual of a homogeneous Boolean algebra. Using the dual Ramsey theorem and a detailed combinatorial analysis of what we call stable collections of subsets of…
We introduce two new iteration games: the game G, which is a strengthening of the weak iteration game, and the game G+, which is somewhat stronger than G but weaker than the full iteration game of length omega_1. For a countable M…
Algebraic properties of $n$-place opening operations on a fixed set are described. Conditions under which a Menger algebra of rank $n$ can be represented by $n$-place opening operations are found.
No convenient internal characterization of spaces that are productively Lindelof is known. Perhaps the best general result known is Alster's internal characterization, under the Continuum Hypothesis, of productively Lindelof spaces which…
We present two zero-sum games modeling situations where one player attacks (or hides in) a finite dimensional nonempty compact set, and the other tries to prevent the attack (or find him). The first game, called patrolling game, corresponds…
In this paper we study the selection principle of closed discrete selection, first researched by Tkachuk in [13] and strengthened by Clontz, Holshouser in [3], in set-open topologies on the space of continuous real-valued functions.…
We use a natural forcing to construct a left-separated topology on an arbitrary cardinal kappa. The resulting left-separated space X_kappa is also 0-dimensional T_2, hereditarily Lindelof, and countably tight. Moreover if kappa is regular…
We study infinite two-player win/lose games $(A,B,W)$ where $A,B$ are finite and $W \subseteq (A \times B)^\omega$. At each round Player 1 and Player 2 concurrently choose one action in $A$ and $B$, respectively. Player 1 wins iff the…
A test space is the set of outcome-sets associated with a collection of experiments. This notion provides a simple mathematical framework for the study of probabilistic theories -- notably, quantum mechanics -- in which one is faced with…
Let $X$ be a compact metrizable space, and let $\Delta$ be a closed set of Borel probability measures on $X$. We study the small boundary property of the pair $(X, \Delta)$. In particular, it is shown that $(X, \Delta)$ has the small…
Some results in C_k-theory are obtained with the use of bornologies. We investigate under which conditions the space of the continuous real functions with the compact-open topology is a productively countably tight space, which yields some…
We discuss a natural topology on powers of a space that is inspired by the Vietoris topology on compact subsets. We then place this topology in context with other product topologies; specifically, we compare this topology with the Tychonoff…