# Pseudoradial spaces and copies of $\omega_1+1$

## TL;DR

This paper explores the relationship between pseudoradial and strongly pseudoradial spaces in compact spaces, demonstrating under certain set-theoretic assumptions that these concepts differ and analyzing conditions for the existence of copies of +1 and .

## Contribution

It constructs a compact pseudoradial space that is not strongly pseudoradial under specific set-theoretic assumptions and investigates conditions for the presence of  copies in such spaces.

## Key findings

- MA + c = _2 implies existence of a compact pseudoradial space not strongly pseudoradial
- Constructed a compact, sequentially compact space with a continuous map to +1 lacking cofinal +1 copies
- PFA implies compact almost radial spaces of radial character  contain many  copies

## Abstract

In this paper we compare the concepts of pseudoradial spaces and the recently defined strongly pseudoradial spaces in the realm of compact spaces. We show that $\mathrm{MA}+\mathfrak{c}=\omega_2$ implies that there is a compact pseudoradial space that is not strongly pseudoradial. We essentially construct a compact, sequentially compact space $X$ and a continuous function $f:X\to\omega_1+1$ in such a way that there is no copy of $\omega_1+1$ in $X$ that maps cofinally under $f$. We also give some conditions that imply the existence of copies of $\omega_1$ in spaces. In particular, $\mathrm{PFA}$ implies that compact almost radial spaces of radial character $\omega_1$ contain many copies of $\omega_1$.