On the cardinality of separable pseudoradial spaces

TL;DR

This paper investigates the maximum possible size of separable pseudoradial spaces, exploring consistency results and bounds under various set-theoretic assumptions, and leaves open the question of existence of larger spaces in ZFC.

Contribution

It provides new consistency results and bounds on the cardinality of separable pseudoradial spaces under different set-theoretic hypotheses.

Findings

Existence of a large countably tight compact SP space under Martin's Axiom.

Bound on the size of countably tight regular SP spaces after adding Cohen reals with a measurable cardinal.

Upper bound on the size of pseudocompact SP spaces with isolated points after Cohen reals addition.

Abstract

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 $\kappa$ is a measurable cardinal then in the forcing extension obtained by adding $\kappa$ many Cohen reals, every countably tight regular SP space has cardinality at most $\mathfrak c$. 3. If $\kappa>\aleph_1$ 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 $\mathfrak c$.