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Spaces of urelements. (English) Zbl 0601.54023

Without the axiom of choice, topologists have new worlds to explore. Here, the author shows readers familiar with Mostowski’s model (described in U. Felgner’s ”Models of ZF-set theory”, Lect. Notes Math. 223 (1971; Zbl 0269.02029), that in this model, a Hausdorff space is anti- anti-compact iff it is a continuous one-to-one image of a Dedekind-finite subset of \(U^{\omega}\), where U is the set of ”urelements”. P. Bankston [Ill. J. Math. 23, 241-252 (1979; Zbl 0405.54003)] called a space anti-anti-compact if each of its infinite subsets has an infinite compact subset. Under the axiom of choice, Hausdorff spaces with that property must be finite.) The author’s results also bear on the interdependence of variants of the axiom of choice.
Reviewer: M.Schroder

MSC:

54D30 Compactness
54A35 Consistency and independence results in general topology
03E35 Consistency and independence results
54A05 Topological spaces and generalizations (closure spaces, etc.)
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References:

[1] P. Bankston , The Total Negation of a Topological Property , Illinois J. Math. , 23 ( 1979 ), pp. 241 - 252 . Article | MR 528560 | Zbl 0405.54003 · Zbl 0405.54003
[2] N. Brunner , \sigma -kompakte Raume , Manuscripta Math ., 38 ( 1982 ), pp. 375 - 379 . Article | Zbl 0504.54004 · Zbl 0504.54004 · doi:10.1007/BF01170932
[3] N. Brunner , Dedekind-Endlichkeit und Wohlordenbarkeit , Monatshefte Math. , 94 ( 1982 ), pp. 9 - 31 . MR 670012 | Zbl 0481.03030 · Zbl 0481.03030 · doi:10.1007/BF01369079
[4] N. Brunner , The Axiom of Choice in Topology , Notre Dame J. Formal Logic , 24 ( 1983 ), pp. 305 - 317 . Article | MR 703494 | Zbl 0487.03022 · Zbl 0487.03022 · doi:10.1305/ndjfl/1093870373
[5] N. Brunner , Positive Functionals and the Axiom of Choice , Rendiconti Sem. Mat. Padova , 71 ( 1983 ) (to appear). Numdam | MR 778328 | Zbl 0554.46007 · Zbl 0554.46007
[6] U. Felgner , Models of ZF Set Theory , Lecture Notes Math. , 223 , Springer , 1971 . MR 351810 | Zbl 0269.02029 · Zbl 0269.02029 · doi:10.1007/BFb0061160
[7] T. Jech , The Axiom of Choice, Studies in Logic 75 , North Holland PC , 1973 . MR 396271 | Zbl 0259.02051 · Zbl 0259.02051
[8] V. Kannan , Countable Compact Spaces , Publ. Math. Debrecen , 21 ( 1974 ), pp. 118 - 120 . MR 355993 | Zbl 0291.54020 · Zbl 0291.54020
[9] J.C. Tong , Almost Continuous Mappings, I , J. Math. M.S. , 6 ( 1983 ), pp. 197 - 199 . MR 689458 | Zbl 0519.54004 · Zbl 0519.54004 · doi:10.1155/S0161171283000198
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