On the weak pseudoradiality of CSC spaces

TL;DR

This paper investigates the properties of compact (weakly) pseudoradial spaces under certain set-theoretic assumptions, showing that in specific forcing extensions, such spaces exhibit particular pseudoradial behaviors and product properties.

Contribution

It establishes new conditions under which compact weakly pseudoradial spaces are pseudoradial and explores product stability of pseudoradiality in the context of set-theoretic assumptions.

Findings

In forcing extensions with property K, all compact sequentially compact spaces are weakly pseudoradial.

Under certain cardinal assumptions, weak pseudoradiality implies pseudoradiality unless the space maps onto a high-dimensional cube.

The product of two compact pseudoradial spaces that is weakly pseudoradial is actually pseudoradial.

Abstract

In this paper, we prove that in forcing extensions by a poset with finally property K over a model of GCH+$\square$, every compact sequentially compact space is weakly pseudoradial. We also prove the following assuming $\mathfrak{s}\leq \aleph_2$: (i) if $X$ is compact weakly pseudoradial, then $X$ is pseudoradial if and only if $X$ cannot be mapped onto $[0,1]^\mathfrak{s}$; (ii) if $X$ and $Y$ are compact pseudoradial spaces such that $X\times Y$ is weakly pseudoradial, then $X\times Y$ is pseudoradial. These results add to the wide variety of partial answers to the question by Gerlits and Nagy of whether the product of two compact pseudoradial spaces is pseudoradial.