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On the Compactness and Countable Compactness of $2^{ℝ}$ in ZF

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Keremedis K., Felouzis E., Tachtsis E.

Bulletin of the Polish Academy of Sciences. Mathematics

2007

tom 55

nr 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 "$2^{ℝ}$ is countably compact" and "$2^{ℝ}$ 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 54

nr 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.

Ograniczanie wyników

2 Bulletin of the Polish Academy of Sciences. Mathematics

2 Felouzis E.

2 Keremedis K.

2 Tachtsis E.

1 2007

1 2006

Wyniki wyszukiwania

On the Compactness and Countable Compactness of $2^{ℝ}$ in ZF

Countable Compact Scattered T₂ Spaces and Weak Forms of AC