Countable compact scattered T2 spaces and weak forms of AC

• Autorzy: Keremedis K. , Felouzis E. , Tachtsis E.
• Języki publikacji: EN

Abstrakty

EN

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.

Słowa kluczowe

EN

axiom of choice; weak axiom of choice; compact spaces; Hausdorff spaces; countable spaces; metrizable spaces; zero-dimensional spaces; scattered spaces; Baire spaces

PL

aksjomat wyboru; słaby aksjomat wyboru; przestrzeń zwarta; przestrzeń Hausdorffa; przestrzeń przeliczalna; przestrzeń metryzowalna; przestrzeń zerowymiarowa; przestrzeń rozproszona; przestrzeń Blaire'a

• Wydawca: Instytut Matematyczny PAN
• Czasopismo: Bulletin of the Polish Academy of Sciences. Mathematics
• Rocznik: 2006
• Tom: Vol. 54, no 1
• Strony: 75--84
• Opis fizyczny: Bibliogr. 7 poz.

Twórcy

autor

Keremedis K.

autor

Felouzis E.

autor

Tachtsis E.

• Department of Mathematics, University of the Aegean, Karlovassi, 83 200, Samos, Greece,

Bibliografia

• [l] C. Good and I. Tree, Continuing horrors of topology without choice, Topology Appl. 63 (1995), 79-90.
• [2] P. Howard and J. E. Rubin, Consequences of the Axiom of Choice, Math. Surveys Monogr. 59, Amer. Math. Soc., Providence RI, 1998.
• [3] K. Keremedis, Some weak forms of the Baire category theorem, Math. Logic Quart. 49 (2003), 369-374.
• [4] K. Keremedis and E. Tachtsis, On sequentially compact subspaces of R without the axiom of choice, Notre Dame J. Formal Logic 44 (2003), 175-184.
• [5] —, —, Countable compact Hausdorff spaces need not be metrizable in ZF, Proc. Amer. Math. Soc., to appear.
• [6] K. Kuratowski, Topology, Vol. I, Academic Press, New York, 1966.
• [7] A. J. Ostaszewski, On countably compact, perfectly normal spaces, J. London Math. Soc. (2) 14 (1976), 505-516