×

zbMATH — the first resource for mathematics

Spaces of Urelements. II. (English) Zbl 0668.54014
Topology thrives without the Axiom of choice. Choosing the Mostowski model [T. J. Jech, ‘The Axiom of Choice’ (1973; Zbl 0259.02051)], the author continues his work on ‘spaces of urelements’ [N. Brunner, Rend. Semin. Mat. Univ. Padova 74, 7-13 (1985; Zbl 0601.54023)]. Here, he explains, “In general, the characterization of a class of spaces which are built up from urelements may lead to rather obscure notions. This is to be expected, because the very existence of these spaces contradicts AC. Interestingly, the class of continuous \(T_ 2\)-images of [the urelement line] L can be described by very harmless looking properties. Our main result states, that these are just the Dedekind-finite Lindelöf \(T_ 2\)-spaces which are hereditarily locally Lindelöf and have at most finitely many isolated points”.
This paper makes no concessions to beginners: readers should know about the Mostowski model and be prepared to decipher a few pieces of dense - if not confusing - symbolism.
Reviewer: M.Schroder
MSC:
54D20 Noncompact covering properties (paracompact, Lindelöf, etc.)
54D30 Compactness
03E60 Determinacy principles
PDF BibTeX XML Cite
Full Text: Numdam EuDML
References:
[1] C.E. Aull , Classification of spaces , Bull. Acad. Polon. Sc. , 15 ( 1967 ), pp. 773 - 778 . MR 225278 | Zbl 0171.43403 · Zbl 0171.43403
[2] P. Bankston , Total negation of a topological property , Illinois J. M. , 23 ( 1979 ), pp. 241 - 252 . Article | MR 528560 | Zbl 0405.54003 · Zbl 0405.54003 · minidml.mathdoc.fr
[3] F. Bernstein : Theorie der trigonometrischen Reihe, Sitzingsber . Leipzig , 60 , ( 1908 ), pp. 325 - 338 . JFM 39.0474.02 · JFM 39.0474.02
[4] A. Blass , A model without ultrafilters , Bull. Acad. Polon. Sc. , 25 ( 1977 ), pp. 329 - 331 . MR 476510 | Zbl 0365.02054 · Zbl 0365.02054
[5] N. Brunner , \sigma -kompakte Raüme , Manuscripta Math ., 38 ( 1982 ), pp. 325 - 379 . Article | Zbl 0504.54004 · Zbl 0504.54004 · doi:10.1007/BF01170932 · eudml:154865
[6] N. Brunner , Lindelöf-Raüme und AC, Anz . Akad. Wiss. Wien , 119 ( 1982 ), pp. 161 - 165 . MR 728812 | Zbl 0529.03030 · Zbl 0529.03030
[7] 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 · eudml:178080
[8] N. Brunner , Spaces of Urelements , Rend. Sem. Mat. Univ. Padova , 74 ( 1985 ), pp. 7 - 13 . Numdam | MR 818710 | Zbl 0601.54023 · Zbl 0601.54023 · numdam:RSMUP_1985__74__7_0 · eudml:108012
[9] J.R. Buddenhagen , Subsets of \omega , Amer. Math. Monthly , 78 ( 1971 ), pp. 536 - 537 .
[10] J.G. Ceder , Resolvable spaces , Fund. Math. , 55 ( 1964 ), pp. 87 - 93 . MR 163279 | Zbl 0139.40401 · Zbl 0139.40401 · eudml:213798
[11] W.W. Comfort , A Stone-Čech theorem without AC , Fund. Math ., 63 ( 1968 ), pp. 97 - 110 . MR 236880 | Zbl 0185.26401 · Zbl 0185.26401 · eudml:214059
[12] E.K. Van Douwen , Horrors of topology , Proc. Amer. Math. Soc. , 95 ( 1985 ), pp. 101 - 105 . MR 796455 | Zbl 0574.03039 · Zbl 0574.03039 · doi:10.2307/2045582
[13] A.G. El’kin , Maximal connected T2-spaces , Mat. Zametki , 26 ( 1979 ), pp. 939 - 948 . MR 567231 | Zbl 0435.54018 · Zbl 0435.54018 · doi:10.1007/BF01142086
[14] R. Engelking , General topology , Polish Sc. Publ. , Warsaw ( 1977 ). MR 500780 | Zbl 0373.54002 · Zbl 0373.54002
[15] U. Felgner , Models of ZF , Springer Lecture Notes 223 ( 1971 ). MR 351810 | Zbl 0269.02029 · Zbl 0269.02029 · doi:10.1007/BFb0061160
[16] M. Gitik , All uncountable cardinals can be singular , Israel J. Math. , 35 ( 1980 ), pp. 61 - 88 . MR 576462 | Zbl 0439.03036 · Zbl 0439.03036 · doi:10.1007/BF02760939
[17] H. Herrlich - V. KANNAN - H. RAJAGOPALAN, Compactness , Proc. Amer. Math. Soc. , 77 ( 1979 ), pp. 421 - 423 . MR 545607 | Zbl 0417.54004 · Zbl 0417.54004 · doi:10.2307/2042197
[18] E. Hewitt , A problem in set theoretic topology , Duke Math. J. , 10 ( 1943 ), pp. 309 - 333 . Article | MR 8692 | Zbl 0060.39407 · Zbl 0060.39407 · doi:10.1215/S0012-7094-43-01029-4 · minidml.mathdoc.fr
[19] T. Jech , The axiom of choice , North Holland Studies in Logic , 75 ( 1973 ). MR 396271 | Zbl 0259.02051 · Zbl 0259.02051
[20] M. Katetov , Disjoint dense subsets , Math. Sbornik , 21 ( 1947 ), pp. 3 - 12 . MR 21679
[21] W.F. Lindgren - P. Fletcher , Problems concerning countably compact spaces, Rocky Mt . J. Math. , 5 ( 1975 ), pp. 95 - 106 . MR 423298 | Zbl 0296.54018 · Zbl 0296.54018 · doi:10.1216/RMJ-1975-5-1-95
[22] N. Lusin - W. Sierpinski , Une décomposition , Compt. Rend. Acad. Paris , 165 ( 1917 ), pp. 422 - 424 . JFM 46.0294.01 · JFM 46.0294.01
[23] P.L. Sharma , Lindelöf property in MI spaces , Illinois J. Math. , 25 ( 1981 ), 644 - 648 . Article | MR 630841 | Zbl 0514.54018 · Zbl 0514.54018 · minidml.mathdoc.fr
[24] E. Specker , Axiomatik der Mengenlehre , Z. Math. Logik und Grundl. Math ., 3 ( 1957 ), pp. 173 - 210 . MR 99297 | Zbl 0079.07605 · Zbl 0079.07605 · doi:10.1002/malq.19570031302
[25] G.T. Whyburn , Connected sets , Trans. Amer. Math. Soc. , 32 ( 1930 ), pp. 926 - 943 . MR 1501572 | JFM 56.1138.03 · JFM 56.1138.03
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.