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Countable Compact Scattered T₂ Spaces and Weak Forms of AC

Kyriakos Keremedis, Evangelos Felouzis, Eleftherios Tachtsis (2006)

Bulletin of the Polish Academy of Sciences. Mathematics

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...

Countable compactness and p -limits

Salvador García-Ferreira, Artur Hideyuki Tomita (2001)

Commentationes Mathematicae Universitatis Carolinae

For M ω * , we say that X is quasi M -compact, if for every f : ω X there is p M such that f ¯ ( p ) X , where f ¯ is the Stone-Čech extension of f . In this context, a space X is countably compact iff X is quasi ω * -compact. If X is quasi M -compact and M is either finite or countable discrete in ω * , then all powers of X are countably compact. Assuming C H , we give an example of a countable subset M ω * and a quasi M -compact space X whose square is not countably compact, and show that in a model of A. Blass and S. Shelah every quasi...

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