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⊂X there is a subset B⊂A such that B⎯⎯⎯⎯∖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.