2020/12/04 by Dow, Alan, Juhasz, Istvan · 1 citation
#54A35 #FOS: Mathematics #General Topology (math.GN)
paper · doi:10.48550/arxiv.2012.02751
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 =ℵ2 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>ℵ1 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≤ℵ2, then there is a 0-dimensional SP space with a countable dense set of isolated points that has cardinal greater than \mathfrak c.