Keremedis, Kyriakos; Tachtsis, Eleftherios Nonconstructive properties of well-ordered T\(_2\) topological spaces. (English) Zbl 0988.54006 Notre Dame J. Formal Logic 40, No. 4, 548-553 (1999). Summary: We show that none of the following statements is provable in Zermelo-Fraenkel set theory (ZF) answering the corresponding open questions from N. Brunner’s paper “The axiom of choice in topology” [Notre Dame J. Formal Logic 24, 305-317 (1983; Zbl 0487.03022): (i) For every \(T_2\) topological space \((X,T)\) if \(X\) is well-ordered, then \(X\) has a well-ordered base,(ii) For every \(T_2\) topological space \((X,T)\), if \(X\) is well-ordered, then there exists a function \(f:X\times W\to T\) such that \(W\) is a well-ordered set and \(f(\{x\}\times W)\) is a neighborhood base at \(x\) for each \(x\in X\),(iii) For every \(T_2\) topological space \((X,T)\), if \(X\) has a well-ordered dense subset, then there exists a function \(f:X \times W\to T\) such that \(W\) is a well-ordered set and \(\{x\}= \bigcap f(\{x\} \times W)\) for each \(x\in X\). MSC: 54A35 Consistency and independence results in general topology 03E25 Axiom of choice and related propositions 03E35 Consistency and independence results 54F05 Linearly ordered topological spaces, generalized ordered spaces, and partially ordered spaces 54G20 Counterexamples in general topology Keywords:well-order; Hausdorff space; Fraenkel-Mostowski permutation models; neighborhood base; Zermelo-Fraenkel set theory Citations:Zbl 0509.03027; Zbl 0487.03022 × Cite Format Result Cite Review PDF Full Text: DOI References: [1] Brunner, N., “The Axiom of Choice in topology,” Notre Dame Journal of Formal Logic, vol. 24 (1983), pp. 305–17. · doi:10.1305/ndjfl/1093870373 [2] Cohen, P. J., Set Theory and the Continuum Hypothesis , Benjamin, New York, 1966. · Zbl 0182.01301 [3] Felgner, U., and T. Jech, “Variants of the axiom of choice in set theory with atoms,” Fundamenta Mathematicæ , vol. 79 (1973), pp. 79–85. · Zbl 0259.02052 [4] Howard, P., and J. E. Rubin, Consequences of the Axiom of Choice , Mathematical Surveys and Monographs, vol. 59, American Mathematical Society, Providence, 1998. · Zbl 0947.03001 [5] Jech, T., The Axiom of Choice , North-Holland, Amsterdam, 1973. · Zbl 0259.02051 [6] Kunen, K., Set Theory, An Introduction to Independence Proofs , North-Holland, Amsterdam, 1983. · Zbl 0534.03026 [7] Monro, G. P., “Independence results concerning Dedekind-finite sets,” Journal of the Australian Mathematical Society , vol. 19 (1975), pp. 35–46. · Zbl 0298.02066 · doi:10.1017/S1446788700023521 [8] Steen, L. A., and J. A. Seebach, Counterexamples in Topology , Dover, New York, 1995. · Zbl 0211.54401 [9] Willard, S., General Topology , Addison-Wesley, Boston, 1968. · Zbl 0205.26601 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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.