特尔加斯基的猜想可能会失败
Telgarsky's conjecture may fail
摘要：
特尔加斯基猜想指出，对于每个$k \in \mathbb N$，存在一个拓扑空间$X_k$，使得在$X_k$上的巴拿赫-马祖尔博弈中，参与者{ \scriptsize NONEMPTY}有一个获胜的$(k+1)$ -战术，但没有获胜的$k$ -战术。我们证明这种说法是一贯错误的。更具体地说，我们证明，假设$\mathsf{GCH}+\square$，如果{ \scriptsize NONEMPTY}有一个获胜的策略，在一个$T_3$空间$X$的巴拿赫-马祖尔游戏，那么她有一个获胜的$2$ -战术。这个证明使用了加尔文的一个编码参数，如果PDMATHAMT有一个具有某些好的属性的PDMATHAMV-基，那么{ \scriptsize NONEMPTY}就能够在她对手的每一个连续的移动对中编码出关于当前移动之前的游戏的所有基本信息。我们的证明表明，在$\mathsf{GCH}+\square$下，每个$T_3$空间都有一个足够好的$π$基，使这种编码策略成为可能。翻译成偏序集的语言，我们真正证明的是$\mathsf{GCH}+\square$隐含了以下陈述，这等价于上面提到的“好”$π$ -基的存在：\emph{Every separative poset $\mathbb P$ with the $κ$-cc contains a dense sub-poset $\mathbb D$ such that $|\{ q \in \mathbb D \,:\, p \text{ extends } q \}| < κ$ for every $p \in \mathbb P$.}我们证明了这个陈述与$\mathsf{ZFC}$无关：当它在$\mathsf{GCH}+\square$下成立时，即使对于ccc偏序集，如果$\mathfrak{b} > \aleph_1$，它也是假的。我们还表明，如果$|\mathbb P| < \aleph_ω$，则\axiom -for- $\mathbb P$是$\mathsf{GCH}$保持低于$|\mathbb P|$的结果。
英文摘要：
Telgársky's conjecture states that for each $k \in \mathbb N$, there is a topological space $X_k$ such that in the Banach-Mazur game on $X_k$, the player {\scriptsize NONEMPTY} has a winning $(k+1)$-tactic but no winning $k$-tactic. We prove that this statement is consistently false. More specifically, we prove, assuming $\mathsf{GCH}+\square$, that if {\scriptsize NONEMPTY} has a winning strategy for the Banach-Mazur game on a $T_3$ space $X$, then she has a winning $2$-tactic. The proof uses a coding argument due to Galvin, whereby if $X$ has a $π$-base with certain nice properties, then {\scriptsize NONEMPTY} is able to encode, in each consecutive pair of her opponent's moves, all essential information about the play of the game before the current move. Our proof shows that under $\mathsf{GCH}+\square$, every $T_3$ space has a sufficiently nice $π$-base that enables this coding strategy. Translated into the language of partially ordered sets, what we really show is that $\mathsf{GCH}+\square$ implies the following statement, which is equivalent to the existence of the "nice'' $π$-bases mentioned above: \emph{Every separative poset $\mathbb P$ with the $κ$-cc contains a dense sub-poset $\mathbb D$ such that $|\{ q \in \mathbb D \,:\, p \text{ extends } q \}| < κ$ for every $p \in \mathbb P$.} We prove that this statement is independent of $\mathsf{ZFC}$: while it holds under $\mathsf{GCH}+\square$, it is false even for ccc posets if $\mathfrak{b} > \aleph_1$. We also show that if $|\mathbb P| < \aleph_ω$, then \axiom-for-$\mathbb P$ is a consequence of $\mathsf{GCH}$ holding below $|\mathbb P|$.
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