13 found
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  1.  12
    Borel's Conjecture in Topological Groups.Fred Galvin & Marion Scheepers - 2013 - Journal of Symbolic Logic 78 (1):168-184.
    We introduce a natural generalization of Borel's Conjecture. For each infinite cardinal number $\kappa$, let ${\sf BC}_{\kappa}$ denote this generalization. Then ${\sf BC}_{\aleph_0}$ is equivalent to the classical Borel conjecture. Assuming the classical Borel conjecture, $\neg{\sf BC}_{\aleph_1}$ is equivalent to the existence of a Kurepa tree of height $\aleph_1$. Using the connection of ${\sf BC}_{\kappa}$ with a generalization of Kurepa's Hypothesis, we obtain the following consistency results: 1. If it is consistent that there is a 1-inaccessible cardinal then it is (...)
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  2.  9
    The Length of Some Diagonalization Games.Marion Scheepers - 1999 - Archive for Mathematical Logic 38 (2):103-122.
    For X a separable metric space and $\alpha$ an infinite ordinal, consider the following three games of length $\alpha$ : In $G^{\alpha}_1$ ONE chooses in inning $\gamma$ an $\omega$ –cover $O_{\gamma}$ of X; TWO responds with a $T_{\gamma}\in O_{\gamma}$ . TWO wins if $\{T_{\gamma}:\gamma<\alpha\}$ is an $\omega$ –cover of X; ONE wins otherwise. In $G^{\alpha}_2$ ONE chooses in inning $\gamma$ a subset $O_{\gamma}$ of ${\sf C}_p(X)$ which has the zero function $\underline{0}$ in its closure, and TWO responds with a function (...)
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  3.  50
    The Algebraic Sum of Sets of Real Numbers with Strong Measure Zero Sets.Andrej Nowik, Marion Scheepers & Tomasz Weiss - 1998 - Journal of Symbolic Logic 63 (1):301-324.
    We prove the following theorems: (1) If X has strong measure zero and if Y has strong first category, then their algebraic sum has property s 0 . (2) If X has Hurewicz's covering property, then it has strong measure zero if, and only if, its algebraic sum with any first category set is a first category set. (3) If X has strong measure zero and Hurewicz's covering property then its algebraic sum with any set in APC ' is a (...)
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  4.  24
    Lebesque Measure Zero Subsets of the Real Line and an Infinite Game.Marion Scheepers - 1996 - Journal of Symbolic Logic 61 (1):246-249.
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  5.  27
    Variations on a Game of Gale (I): Coding Strategies.Marion Scheepers - 1993 - Journal of Symbolic Logic 58 (3):1035-1043.
    We consider an infinite two-person game. The second player has a winning perfect information strategy; we show that this player has a winning strategy which depends on substantially less information. The game studied here is a variation on a game of Gale.
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  6.  6
    Baire Spaces and Infinite Games.Fred Galvin & Marion Scheepers - 2016 - Archive for Mathematical Logic 55 (1-2):85-104.
    It is well known that if the nonempty player of the Banach–Mazur game has a winning strategy on a space, then that space is Baire in all powers even in the box product topology. The converse of this implication may also be true: We know of no consistency result to the contrary. In this paper we establish the consistency of the converse relative to the consistency of the existence of a proper class of measurable cardinals.
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  7.  32
    Combinatorial Properties of Filters and Open Covers for Sets of Real Numbers.Claude Laflamme & Marion Scheepers - 1999 - Journal of Symbolic Logic 64 (3):1243-1260.
    We analyze combinatorial properties of open covers of sets of real numbers by using filters on the natural numbers. In fact, the goal of this paper is to characterize known properties related to ω-covers of the space in terms of combinatorial properties of filters associated with these ω-covers. As an example, we show that all finite powers of a set R of real numbers have the covering property of Menger if, and only if, each filter on ω associated with its (...)
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  8.  8
    Concerning N-Tactics in the Countable-Finite Game.Marion Scheepers - 1991 - Journal of Symbolic Logic 56 (3):786-794.
  9.  3
    Concerning N-Tactics in the Countable-Finite Game.Marion Scheepers - 1991 - Journal of Symbolic Logic 56 (3):786.
  10.  55
    Finite Powers of Strong Measure Zero Sets.Marion Scheepers - 1999 - Journal of Symbolic Logic 64 (3):1295-1306.
    In a previous paper-[17]-we characterized strong measure zero sets of reals in terms of a Ramseyan partition relation on certain subspaces of the Alexandroff duplicate of the unit interval. This framework gave only indirect access to the relevant sets of real numbers. We now work more directly with the sets in question, and since it costs little in additional technicalities, we consider the more general context of metric spaces and prove: 1. If a metric space has a covering property of (...)
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  11.  28
    Meager Nowhere-Dense Games (IV): N-Tactics.Marion Scheepers - 1994 - Journal of Symbolic Logic 59 (2):603-605.
    We consider the infinite game where player ONE chooses terms of a strictly increasing sequence of first category subsets of a space and TWO chooses nowhere dense sets. If after ω innings TWO's nowhere dense sets cover ONE's first category sets, then TWO wins. We prove a theorem which implies for the real line: If TWO has a winning strategy which depends on the most recent n moves of ONE only, then TWO has a winning strategy depending on the most (...)
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  12. Rothberger's Property and Partition Relations.Marion Scheepers - 1997 - Journal of Symbolic Logic 62 (3):976-980.
  13.  16
    Variations on a Game of Gale (III): Remainder Strategies.Marion Scheepers & William Weiss - 1997 - Journal of Symbolic Logic 62 (4):1253-1264.