Cardinal estimates involving the weak Lindelöf game (Q2240599)

All hypothesized spaces are Hausdorff, and \(\kappa\) denotes an infinite cardinal. The authors discuss and answer or partially answer a number of previously raised open questions primarily concerning bounds on \(|X|\), where a space \(X\) satisfies various cover and character conditions. Some of these questions are: (a) For a regular space \(X\), is \(|X|\leq 2^{wL(X)\cdot\chi(X)}\)? This question was first raised by Bell, Ginsburg and Woods in [\textit{M. Bell} et al., Pac. J. Math. 79, 37--45 (1979; Zbl 0367.54003)], who proved the answer to be ``yes'' for a normal space \(X\). (b) If \(X\) is a first countable regular space where Player II has a winning strategy in \(G^{\omega_1}_{fin}(\mathcal O, \mathcal O_D)\), is \(|X|\leq2^{\aleph_0}\)? (c) Is there an example of a weakly Lindelöf space \(X\) where Player II does not have a winning strategy in \(G_{1}^{\omega_1}(\mathcal O,\mathcal O_D)\)? Here, \(\chi(X)\) denotes the chracter of \(X\), and \(wL(X)\) denotes the \textit{weak Lindelöf number of \(X\)} and is defined to be the least infinite cardinal \(\kappa\) such that for every open cover \(\mathcal U\) of \(X\) there is a subfamily \(\mathcal V\subset\mathcal U\) of cardinality \(\leq\kappa\) such that \(X=\overline{\bigcup\mathcal V}\). The authors define ``a game \(G_{1}^{\kappa}(\mathcal O,\mathcal O_D)\) (resp., \(G^{\kappa}_{fin}(\mathcal O,\mathcal O_D)\)) as follows: at inning \(\alpha<\kappa\), Player I plays an open cover \(\mathcal{U}_\alpha\) of \(X\) and Player II plays an open set \(U_\alpha\in\mathcal U_\alpha\) (resp., a finite subcollection \(\mathcal F_\alpha\subset\mathcal U_\alpha)\). Player II wins if the collection \(\{U_\alpha:\alpha<\kappa\}\) has a dense union in \(X\) (resp., if the collection \(\bigcup\{ \mathcal{F}_\alpha:\alpha<\kappa\}\) has a dense union in \(X\)).'' They note that if Player II has a winning strategy in \(G_{1}^{\kappa}(\mathcal O,\mathcal O_D)\) on \(X\), then Player II has a winning strategy in \(G^{\kappa}_{fin}(\mathcal O,\mathcal O_D)\) on \(X\), and hence \(wL(X)\leq\kappa\). In addition, they denote by \(\mathcal O_K\) the family of all open covers \(\mathcal O\) of a space \(X\) such that for every compact set \(K\subset X\) there is a set \(O\in\mathcal O\) such that \(K\subset O\), and they recall that by a \textit{Urysohn space} one means a space in which any two distinct points can be separated by disjoint closed neighborhoods of those points. Some of the main results obtained are the following. Theorem 3: If \(X\) is a Urysohn space such that \(\chi(X)<\kappa\) and Player II has a winning strategy in \(G_{1}^{\kappa}(\mathcal O,\mathcal O_D)\), then \(|X|\leq2^{<\kappa}\). Improving a previously published analogous theorem requiring the space \(X\) to be regular (see [\textit{A. Bella} and \textit{S. Spadaro}, Houston J. Math. 41, No. 3, 1063--1077 (2015; Zbl 1343.54003)]), Theorem 3 implies Corollary 4: For any Urysohn first countable space \(X\) where Player II has a winning strategy in \(G_{1}^{\omega_1}(\mathcal O,\mathcal O_D)\), one has \(|X|\leq2^{\aleph_0}\). Illustrating one way in which Theorem 3 cannot be strengthened, they show (using an example in [\textit{M. Bell} et al., loc. cit.]) that there are first countable Hausdorff spaces of arbitrarily large cardinality, in each of which Player II has a winning strategy in \(G_{1}^{\omega}(\mathcal O,\mathcal O_D)\). Then the authors present Theorem 6, which differs from Corollary 4 only in that the hypothesis word ``Urysohn'' has been replaced by ``countably compact Hausdorff,'' and they provide a proof of Theorem 6. While that proof is interesting and uses elementary submodels, some readers may notice that it is not needed, and Theorem 6 is actually an immediate corollary of Corollary 4, for every first countable countably compact Hausdorff space is regular (e.g., as shown in 3.5 on p. 230 of [\textit{J. Dugundji}, Topology. Boston: Allyn and Bacon, Inc (1966; Zbl 0144.21501)]), and every regular \(T_1\)-space is Urysohn. Following Theorem 6 are several lemmas and theorems partially answering question (b). Theorem 9: If \(X\) is a regular first countable space such that Player II has a winning strategy in \(G_{1}^{\omega_1}(\mathcal{O}_K,\mathcal O_D)\), then \(|X|\leq2^{\aleph_0}\). Theorem 13: If \(X\) is a sequential space with \(\chi(X)\leq2^\omega\) such that Player II has a monotone winning strategy for \(G^{\omega_1}_{fin}(\mathcal O, \mathcal O_D)\), then \(|X|\leq2^\omega.\) In the final section, answering question (c), they modify an approach used by \textit{R. R. Dias} and \textit{F. D. Tall} in [Topology Appl. 160, No. 18, 2411--2426 (2013; Zbl 1295.54026)] and provide very nice proofs that \(2^{\omega_2}\), with the lexicographic order topology, is a compact almost \(P_{\omega_2}\)-space, and for any compact almost \(P_{\omega_2}\)-space without isolated points, Player I has a winning strategy in \(G^{\omega_1}_{1}(\mathcal O, \mathcal O_D)\).