We show that: (1) It is provable in ZF (i.e., Zermelo-Fraenkel set theory minus the Axiom of Choice AC) that every compact scattered T₂ topological space is zero-dimensional. (2) If every countable union of countable sets of reals is countable, then a countable compact T₂ space is scattered iff it is metrizable. (3) If the real line ℝ can be expressed as a well-ordered union of well-orderable sets, then every countable compact zero-dimensional T₂ space is scattered. (4) It is not provable in ZF+¬AC that there exists a countable compact T₂ space which is dense-in-itself.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-ba54-1-7,
author = {Kyriakos Keremedis and Evangelos Felouzis and Eleftherios Tachtsis},
title = {Countable Compact Scattered T2 Spaces and Weak Forms of AC},
journal = {Bulletin of the Polish Academy of Sciences. Mathematics},
volume = {54},
year = {2006},
pages = {75-84},
zbl = {1115.03066},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-ba54-1-7}
}
Kyriakos Keremedis; Evangelos Felouzis; Eleftherios Tachtsis. Countable Compact Scattered T₂ Spaces and Weak Forms of AC. Bulletin of the Polish Academy of Sciences. Mathematics, Tome 54 (2006) pp. 75-84. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-ba54-1-7/