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arXiv 2012.027512020-12-08math.GN

On the cardinality of separable pseudoradial spaces

Alan Dow, Istvan Juhasz

英文摘要：

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 $\mathfrak c$.

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