论文标题
在$ \ mathbf {zf} $中，豪斯多夫空间和布尔代数的塔楼中的蜂窝家族
Denumerable cellular families in Hausdorff spaces and towers of Boolean algebras in $\mathbf{ZF}$
论文作者
论文摘要
拓扑空间$ \ mathbf {x} $的不可算的蜂窝家族是$ \ mathbf {x} $的成对脱节非空的开放集的无限计数集合。事实证明，以下语句在$ \ mathbf {zf} $中相当于： （i）对于每个无限套装$ x，[x]^{<ω} \ mathbf {\} $都有一个不可算的子集。 （ii）每个无限的$ 0 $维二维的豪斯多夫太空都承认一个不可或缺的蜂窝家族。 还证明（i）暗示以下内容： （iii）每个无限的Hausdorff Baire空间都有一个不可忽视的蜂窝家族。 除其他结果外，还以$ \ mathbf {zf} $： （iv）$ \ mathbb {r} $的每一个可数的非置亚集集合具有一个选择功能，如果每个无限的第二个可容纳的hausdorff space $ \ mathbf {x} $，它认为每个基础的$ \ mathbf {x} $的每个基础都包含一个具有质量的蜂窝家族的$ \ m m i \ \ x} $ {x x x = （v）如果每个Cantor Cube都是伪造的，则每个非空数有限套件的可计数集合都有一个选择函数。 （vi）如果所有cantor立方体都是次核管，则（i）持有。 此外，在其他形式中独立于$ \ mathbf {Zf} $，这是一项部分Kinna-Wagner的选择原理，适用于有限套装的有限族家族的可数工会。事实证明，如果这种新的选择原理和（i）保持，那么每个无限的布尔代数都有一个塔，每个无限的Hausdorff空间都有一个不可忽视的蜂窝家族。
A denumerable cellular family of a topological space $\mathbf{X}$ is an infinitely countable collection of pairwise disjoint non-empty open sets of $ \mathbf{X}$. It is proved that the following statements are equivalent in $\mathbf{ZF}$: (i) For every infinite set $X,[X]^{<ω}\mathbf{\ }$has a denumerable subset. (ii) Every infinite $0$-dimensional Hausdorff space admits a denumerable cellular family. It is also proved that (i) implies the following: (iii) Every infinite Hausdorff Baire space has a denumerable cellular family. Among other results, the following theorems are also proved in $\mathbf{ZF}$: (iv) Every countable collection of non-empty subsets of $\mathbb{R}$ has a choice function iff, for every infinite second-countable Hausdorff space $ \mathbf{X}$, it holds that every base of $\mathbf{X}$ contains a denumerable cellular family of $\mathbf{X}$. (v) If every Cantor cube is pseudocompact, then every non-empty countable collection of non-empty finite sets has a choice function. (vi) If all Cantor cubes are countably paracompact, then (i) holds. Moreover, among other forms independent of $\mathbf{ZF}$, a partial Kinna-Wagner selection principle for families expressible as countable unions of finite families of finite sets is introduced. It is proved that if this new selection principle and (i) hold, then every infinite Boolean algebra has a tower and every infinite Hausdorff space has a denumerable cellular family.
