Countable Compact Scattered T₂ Spaces and Weak Forms of AC

Kyriakos Keremedis, Evangelos Felouzis, and Eleftherios Tachtsis. "Countable Compact Scattered T₂ Spaces and Weak Forms of AC." Bulletin of the Polish Academy of Sciences. Mathematics 54.1 (2006): 75-84. <http://eudml.org/doc/280709>.

@article{KyriakosKeremedis2006,
abstract = { 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. },
author = {Kyriakos Keremedis, Evangelos Felouzis, Eleftherios Tachtsis},
journal = {Bulletin of the Polish Academy of Sciences. Mathematics},
keywords = {axiom of choice; weak forms of axiom of choice; compact spaces; Hausdorff spaces; countable spaces; metrizable spaces; zero-dimensional spaces; scattered spaces; Baire spaces},
language = {eng},
number = {1},
pages = {75-84},
title = {Countable Compact Scattered T₂ Spaces and Weak Forms of AC},
url = {http://eudml.org/doc/280709},
volume = {54},
year = {2006},
}

TY - JOUR
AU - Kyriakos Keremedis
AU - Evangelos Felouzis
AU - Eleftherios Tachtsis
TI - Countable Compact Scattered T₂ Spaces and Weak Forms of AC
JO - Bulletin of the Polish Academy of Sciences. Mathematics
PY - 2006
VL - 54
IS - 1
SP - 75
EP - 84
AB - 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.
LA - eng
KW - axiom of choice; weak forms of axiom of choice; compact spaces; Hausdorff spaces; countable spaces; metrizable spaces; zero-dimensional spaces; scattered spaces; Baire spaces
UR - http://eudml.org/doc/280709
ER -