Related papers: Selection Principles and Baire spaces

I provide simplified proofs for each of the following fundamental theorems regarding selection principles: 1. The Quasinormal Convergence Theorem, due to the author and Zdomskyy, asserting that a certain, important property of the space of…

We study the relation between the Hurewicz and Menger properties of filters considered topologically as subspaces of P(\omega) with the Cantor set topology.

By adapting techniques of Arhangel'skii, Barman, and Dow, we may equate the existence of perfect-information, Markov, and tactical strategies between two interesting selection games. These results shed some light on Gruenhage's question…

Let $X$ be a Banach space with an unconditional finite-dimensional Schauder decomposition $(E_n)$. We consider the general problem of characterizing conditions under which one can construct an unconditional basis for $X$ by forming an…

We answer in the affirmative the surprisingly difficult questions: If a complex Banach space possesses a real predual X, then is X a complex Banach space? If a complex Banach space possesses a real predual, then does it have a complex…

We explore a version of the minimax theorem for two-person win-lose games with infinitely many pure strategies. In the countable case, we give a combinatorial condition on the game which implies the minimax property. In the general case, we…

We create a new two-player game on the Sperner Triangle based on Sperner's lemma. Our game has simple rules and several desirable properties. First, the game is always certain to have a winner. Second, like many other interesting games such…

We give an elementary proof that in a Borel family of games, the set of games for which player II has a winning strategy is Baire measurable, universally measurable, and completely Ramsey in the case where $X = [\mathbb{N}]^{\aleph_0}$.

Necessary and sufficient conditions for a separable Banach space to be a dual space are proved. Some applications are discussed

We develop a theory of combinatorial games that is appropriate for describing positions in Hex and other monotone set coloring games. We consider two natural conditions on such games: a game is monotone if all moves available to both…

Muller games are played by two players moving a token along a graph; the winner is determined by the set of vertices that occur infinitely often. The central algorithmic problem is to compute the winning regions for the players. Different…

We prove that the kernel of a quotient operator from an $\mathcal L_1$-space onto a Banach space $X$ with the Bounded Approximation Property (BAP) has the BAP. This completes earlier results of Lusky --case $\ell_1$-- and Figiel, Johnson…

We prove a separable reduction theorem for sigma-porosity of Suslin sets. In particular, if A is a Suslin subset in a Banach space X, then each separable subspace of X can be enlarged to a separable subspace V such that A is sigma-porous in…

We study productive properties of gamma spaces, and their relation to other, classic and modern, selective covering properties. Among other things, we prove the following results: 1. Solving a problem of F. Jordan, we show that for every…

A topological space $X$ is Baire if the intersection of any sequence of open dense subsets of $X$ is dense in $X$. One of the interesting problems for the space of Baire functions is the Banakh-Gabriyelyan problem: Let $\alpha$ be a…

We are concerned with the problem of witnessing the Baire property of the Borel and the projective sets (assuming determinacy) through a sufficiently definable function in the codes. We prove that in the case of projective sets it is…

We want to give a construction as simple as possible of a Borel subset of a product of two Polish spaces. This introduces the notion of potential Wadge class. Among other things, we study the non-potentially closed sets, by proving…

A topological space $X$ is Baire if the Baire Category Theorem holds for $X$, i.e., the intersection of any sequence of open dense subsets of $X$ is dense in $X$. In this paper, we have obtained that the space $B^{st}_1(X)$ of pointwise…

Given a free ideal J of subsets of a set X, we consider games where player ONE plays an increasing sequence of elements of the sigma completion of J, and TWO tries to cover the union of this sequence by playing one set at a time from J. We…

Given a topological property $\mathcal{P}$, a space $X$ is called star-$\mathcal{P}$ if for any open cover $\mathcal{U}$ of the space $X$, there exists a set $Y\subseteq X$ with property $\mathcal{P}$ such that $St(Y,\mathcal{U})=X$; the…