Countable Compact Scattered T$_{2}$ Spaces and Weak Forms of AC

Volume 54 / 2006

Kyriakos Keremedis, Evangelos Felouzis, Eleftherios Tachtsis 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.

Authors

  • Kyriakos KeremedisDepartment of Mathematics
    University of the Aegean
    Karlovassi, 83 200, Samos, Greece
    e-mail
  • Evangelos FelouzisDepartment of Mathematics
    University of the Aegean
    Karlovassi, 83 200, Samos, Greece
    e-mail
  • Eleftherios TachtsisDepartment of Statistics and
    Actuarial-Financial Mathematics
    University of the Aegean
    Karlovassi, 83 200, Samos, Greece
    e-mail

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