Countably compact weakly Whyburn spaces

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.