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2 Bulletin of the Polish Academy of Sciences. Mathematics

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On the compactness and countable compactness of 2R in ZF

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Keremedis K. , Felouzis E. , Tachtsis E.

Bulletin of the Polish Academy of Sciences. Mathematics

2007

tom Vol. 55, no 4

293-302

In the framework of ZF (Zermelo-Fraenkel set theory without the Axiom of Choice) we provide topological and Boolean-algebraic characterizations of the statements "2R is countably compact" and "2R is compact".

Countable compact scattered T2 spaces and weak forms of AC

100%

Keremedis K. , Felouzis E. , Tachtsis E.

Bulletin of the Polish Academy of Sciences. Mathematics

2006

tom Vol. 54, no 1

75-84

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.