Countable Compact Scattered T$_{2}$ 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.
