Article

 Title: Disasters in metric topology without choice (English) Author: Tachtsis, Eleftherios Language: English Journal: Commentationes Mathematicae Universitatis Carolinae ISSN: 0010-2628 (print) ISSN: 1213-7243 (online) Volume: 43 Issue: 1 Year: 2002 Pages: 165-174 . Category: math . Summary: We show that it is consistent with ZF that there is a dense-in-itself compact metric space $(X,d)$ which has the countable chain condition (ccc), but $X$ is neither separable nor second countable. It is also shown that $X$ has an open dense subspace which is not paracompact and that in ZF the Principle of Dependent Choice, DC, does not imply {\it the disjoint union of metrizable spaces is normal\/}. (English) Keyword: Axiom of Choice Keyword: Axiom of Multiple Choice Keyword: Principle of Dependent Choice Keyword: Ordering Principle Keyword: metric spaces Keyword: separable metric spaces Keyword: second countable metric spaces Keyword: paracompact spaces Keyword: compact T$_2$ spaces Keyword: ccc spaces. MSC: 03E25 MSC: 54A35 MSC: 54D20 MSC: 54E35 MSC: 54E45 MSC: 54F05 idZBL: Zbl 1072.03030 idMR: MR1903316 . Date available: 2009-01-08T19:20:40Z Last updated: 2012-04-30 Stable URL: http://hdl.handle.net/10338.dmlcz/119309 . Reference: [1] Cohen P.J.: Set Theory and the Continuum Hypothesis.Benjamin, 1966. Zbl 0182.01401, MR 0232676 Reference: [2] van Douwen E.K.: Horrors of topology without AC: a non normal orderable space.Proc. Amer. Math. Soc. 95 (1985), 101-105. MR 0796455 Reference: [3] Good C., Tree I.J.: Continuing horrors of topology without choice.Topology Appl. 63 (1995), 79-90. Zbl 0822.54001, MR 1328621 Reference: [4] Good C., Tree I.J., Watson W.S.: On Stone's theorem and the axiom of choice.Proc. Amer. Math. Soc. 126 (1998), 1211-1218. Zbl 0893.54016, MR 1425122 Reference: [5] Herrlich H., Steprāns J.: Maximal filters, continuity and choice principles.Quaestiones Math. 20 (1997), 697-705. MR 1625478 Reference: [6] Herrlich H., Strecker G.E.: When is $\Bbb N$ Lindelöf?.Comment. Math. Univ. Carolinae 38.3 (1997), 553-556. Zbl 0938.54008, MR 1485075 Reference: [7] Howard P., Keremedis K., Rubin H., Rubin J.E.: Disjoint unions of topological spaces and choice.Math. Logic Quart. 44 (1998), 493-508. Zbl 0922.03069, MR 1654348 Reference: [8] Howard P., Keremedis K., Rubin J.E., Stanley A.: Paracompactness of metric spaces and the axiom of multiple choice.Math. Logic Quart. 46 (2000). Zbl 0993.03059, MR 1755811 Reference: [9] Howard P., Keremedis K., Rubin J.E., Stanley A., Tachtsis E.: Non-constructive properties of the real numbers.Math. Logic Quart. 47 (2001), 423-431. MR 1847458 Reference: [10] Howard P., Rubin J.E.: Consequences of the Axiom of Choice.Math. Surveys and Monographs 59, Amer. Math. Soc., Providence R.I., 1998. Zbl 0947.03001, MR 1637107 Reference: [11] Jech T.: The Axiom of Choice.North-Holland, Amsterdam, 1973. Zbl 0259.02052, MR 0396271 Reference: [12] Keremedis K.: Disasters in topology without the axiom of choice.Arch. Math. Logic, to appear. Zbl 1027.03040, MR 1867681 Reference: [13] Keremedis K.: Countable disjoint unions in topology and some weak forms of the axiom of choice.Arch. Math. Logic, submitted. Reference: [14] Keremedis K., Tachtsis E.: Compact metric spaces and weak forms of the axiom of choice.Math. Logic Quart. 47 (2001), 117-128. Zbl 0968.03057, MR 1808950 Reference: [15] Keremedis K., Tachtsis E.: On Lindelöf metric spaces and weak forms of the axiom of choice.Math. Logic Quart. 46 (2000), 35-44. Zbl 0952.03060, MR 1736648 Reference: [16] Kunen K.: Set Theory, An Introduction to Independence Proofs.North-Holland, Amsterdam, 1983. Zbl 0534.03026, MR 0756630 Reference: [17] Willard S.: General Topology.Addison-Wesley Publ. Co., Reading, MA, 1968. Zbl 1052.54001, MR 2048350 .

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