Countably compact weakly Whyburn spaces

TL;DR

This paper investigates the properties of countably compact weakly Whyburn spaces, establishing their characteristics under Urysohn conditions, and provides examples and conditions related to pseudoradiality.

Contribution

It proves that countably compact Urysohn spaces of small cardinality are weakly Whyburn and constructs a non-pseudoradial example, addressing open questions.

Findings

Countably compact Urysohn spaces of size less than continuum are weakly Whyburn.

Conditions are identified under which weakly Whyburn spaces are pseudoradial.

A countably compact weakly Whyburn regular space that is not pseudoradial is constructed.

Abstract

The weak Whyburn property is a generalization of the classical sequential property that has been studied by many authors. A space $X$ is weakly Whyburn if for every non-closed set $A \subset X$ there is a subset $B \subset A$ such that $\overline{B} \setminus A$ is a singleton. We prove that every countably compact Urysohn space of cardinality smaller than the continuum is weakly Whyburn and show that, consistently, the Urysohn assumption is essential. We also give conditions for a (countably compact) weak Whyburn space to be pseudoradial and construct a countably compact weakly Whyburn non-pseudoradial regular space, which solves a question asked by Bella in private communication.