Infinite games and chain conditions

Abstract

Abstract. We apply the theory of infinite two-person games to two well-known problems in topology: Suslin's Problem and Arhangel'skii's problem on G δ covers of compact spaces. More specifically, we prove results of which the following two are special cases: 1) every linearly ordered topological space satisfying the game-theoretic version of the countable chain condition is separable and 2) in every compact space satisfying the game-theoretic version of the weak Lindelöf property, every cover by G δ sets has a continuum-sized subcollection whose union is G δ -dense. IntroductionInfinite games have been exploited in recent years to give partial answers to various important problems in General Topology, including van Douwen's D-space problem (see [1]), Arhangel'skii's problem on the cardinality of Lindelöf spaces with points G δ (see [14], [2]) and Bell, Ginsburg and Woods's problem on the cardinality of weakly Lindelöf first-countable regular spaces (see [6], [3], [4]). We use them to give partial ZFC answers to Suslin's Problem and Arhangel'skii's question of whether in every compact space, a cover by G δ sets has a continuum-sized subfamily with a G δ -dense union.It was already known to Cantor that the real line is the unique complete dense linear order without endpoints which is separable. In the first issue of Fundamenta Mathematicae, Suslin asked whether separability could be replaced with the countable chain condition in this result. Any counterexample to this assertion came to be known as a Suslin Line. The problem turned out to be independent of the usual axioms of ZFC: under MA ω 1 there are no Suslin Lines, and thus the answer to Suslin's question is yes. However, Suslin Lines can be found in certain models of set theory, for example under V = L. Various mathematicians wondered whether there was a natural strengthening of the ccc which implied a positive answer to Suslin's Problem. For example, Knaster proved that every ordered continuum with the Knaster property is separable and Shapirovskii proved that every compact space with countable tightness and Shanin's condition is separable (see [15]).2000 Mathematics Subject Classification. Primary: 54A25, 91A44; Secondary: 54F05, 54G10.

$G_δ$ covers of compact spaces

“…For example, Juhász proved in [15] that wL(X δ ) ≤ 2 ℵ 0 for every compact ccc space X, using the Erdös-Rado theorem. The first-named author generalized this in [25] to prove that wL(X δ ) ≤ 2 ℵ 0 for all spaces X such that player II has a winning strategy in G ω 1 (O, O D ), that is the two-player game in ω 1 many innings where at inning α < ω 1 , player one chooses a maximal family of non-empty pairwise disjoint open sets U α and player two chooses U α ∈ U α and player two wins if {U α : α < ω 1 } is dense in X. In [11], Fleischmann and Williams proved that L(X δ ) ≤ 2 ℵ 0 for every compact linearly ordered space X and Pytkeev proved in [23] that L(X δ ) ≤ 2 ℵ 0 for every compact countably tight space X. Carlson, Porter and Ridderbos generalized the latter result in [7] by proving that L(X δ ) ≤ 2 F (X)t(X)L(X) , where F (X) is the supremum of the cardinalities of the free sequences in X.…”

“…It is natural to ask what properties are preserved when passing from X to X κ and whether cardinal invariants of X κ can be bound in terms of cardinal invariants of X. There has been a fair amount of work in the past on this general question, especially for chain conditions and covering properties (see for example [20], [17], [15], [11], [12], [25]). The aim of this section is to present some preservation results that are related to Arhangel'skii's problems mentioned in the introduction.…”

A Common Extension of Arhangel’skĭ’s Theorem and the Hajnal–Juhász Inequality

“…We call X κ , the G κ -modification of X; in case κ = ω we speak of the G δ -modification of X and we often use the symbol X δ instead. This construction has been extensively studied in the literature; various authors have tried to bound the cardinal functions of X κ in terms of their values on X (see, for example [8], [12], [16], [17], [18]) and results of this kind have found applications to other topics in topology, like the estimation of the cardinality of compact homogeneous spaces (see [5], [8], [9] and [18]).…”