Countable Compact Scattered T₂Spaces and Weak Forms of AC

Abstract

We show that: (1) It is provable in $\textbf{ZF}$ (i.e., Zermelo&#8211;Fraenkel set theory minus the Axiom of Choice $\textbf{AC}$) that <i>every compact scattered</i> T$_{2}$ <i>topological space is zero-dimensional</i>. (2) If <i>every countab