Countable Compact Scattered T Spaces and Weak Forms of AC
Volume 54 / 2006
Bulletin Polish Acad. Sci. Math. 54 (2006), 75-84
MSC: 03E25, 03E35, 54A35, 54D10, 54D30, 54D80, 54E35, 54E52, 54G12.
DOI: 10.4064/ba54-1-7
Abstract
We show that:
(1) It is provable in \textbf{ZF} (i.e., Zermelo–Fraenkel set theory minus the Axiom of Choice \textbf{AC}) that every compact scattered T_{2} topological space is zero-dimensional.
(2) If every countable union of countable sets of reals is countable, then a countable compact T_{2} space is scattered iff it is metrizable.(3) If the real line \mathbb{R} can be expressed as a well-ordered union of well-orderable sets, then every countable compact zero-dimensional T_{2} space is scattered.
(4) It is not provable in \textbf{ZF}+\neg\textbf{AC} that there exists a countable compact T_{2} space which is dense-in-itself.
