论文标题
关于可分离假空间的基数
On the cardinality of separable pseudoradial spaces
论文作者
论文摘要
本文的目的是考虑有关各种可分离伪层（简而言之：SP）空间的最大基数的问题。这里最有趣的问题是，在ZFC中，是否存在大于$ \ Mathfrak C $的常规（或仅是Hausdorff）的SP。虽然这个问题是打开的，但我们确定了我们列出的许多非平凡结果：1。它与Martin的Axiom和$ \ Mathfrak C = \ Aleph_2 $一致，因为有一个非常紧密且紧凑的心脏紧缩$ 2^{\ Mathfrak C} $。 2。如果$κ$是可测量的基础主教，那么在通过添加$κ$许多Cohen Reals获得的强迫延伸中，每个计的常规SP空间最多都具有$ \ Mathfrak C $。 3。如果$κ> \ aleph_1 $ cohen Reals被添加到GCH型号中，则在扩展名中，每个pseudocompact s s sp space sp space sp space s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s sp sp sp sp sp sp sp sp c $的可数值可计数最多，最多具有$ \ mathfrak c $。 4。如果$ \ Mathfrak C \ Leq \ Aleph_2 $，则有一个0维的SP空间，具有可计数的密集隔离点，其基数大于$ \ Mathfrak C $。
The aim of this paper is to consider questions concerning the possible maximum cardinality of various separable pseudoradial (in short: SP) spaces. The most intriguing question here is if there is, in ZFC, a regular (or just Hausdorff) SP of cardinality greater than $\mathfrak c$. While this question is left open, we establish a number of non-trivial results that we list: 1. It is consistent with Martin's Axiom and $\mathfrak c =\aleph_2$ that there is a countably tight and compact SP of cardinality $2^{\mathfrak c}$. 2. If $κ$ is a measurable cardinal then in the forcing extension obtained by adding $κ$ many Cohen reals, every countably tight regular SP space has cardinality at most $\mathfrak c$. 3. If $κ>\aleph_1$ Cohen reals are added to a model of GCH, then in the extension every pseudocompact SP space with a countable dense set of isolated points has cardinality at most $\mathfrak c$. 4. If $\mathfrak c\leq\aleph_2$, then there is a 0-dimensional SP space with a countable dense set of isolated points that has cardinal greater than $\mathfrak c$.
