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

Kyriakos Keremedis¹ ; Evangelos Felouzis¹ ; Eleftherios Tachtsis²

¹ Department of Mathematics University of the Aegean Karlovassi, 83 200, Samos, Greece
² Department of Statistics and Actuarial-Financial Mathematics University of the Aegean Karlovassi, 83 200, Samos, Greece

Bulletin of the Polish Academy of Sciences. Mathematics, Tome 54 (2006) no. 1, pp. 75-84.

Résumé

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.

Détail

DOI : 10.4064/ba54-1-7

Keywords: provable textbf zermelo fraenkel set theory minus axiom choice textbf every compact scattered topological space zero dimensional every countable union countable sets reals countable countable compact space scattered metrizable real line mathbb expressed well ordered union well orderable sets every countable compact zero dimensional space scattered provable textbf neg textbf there exists countable compact space which dense in itself