×

zbMATH — the first resource for mathematics

Disjoint unions of topological spaces and choice. (English) Zbl 0922.03069
This paper continues the authors’ paper “Versions of normality and some weak forms of the axiom of choice” [Math. Log. Q. 44, 367-382 (1998; Zbl 0911.03027)] about the set-theoretical strength of theorems which assert that disjoint unions of topological spaces inherit properties from the summands. [Recall for example: The axiom of multiple choice is equivalent to “disjoint unions of normal spaces are normal”.] The main result of the present paper is Theorem 2: The axiom of multiple choice is equivalent to “disjoint unions of paracompact spaces are normal”. Another interesting result is Example 3: The “ordered Mostowski permutation model” satisfies “disjoint unions of metrizable spaces are metrizable”.
Reviewer: N.Brunner (Wien)

MSC:
03E25 Axiom of choice and related propositions
54B15 Quotient spaces, decompositions in general topology
54D10 Lower separation axioms (\(T_0\)–\(T_3\), etc.)
54D20 Noncompact covering properties (paracompact, Lindelöf, etc.)
54D15 Higher separation axioms (completely regular, normal, perfectly or collectionwise normal, etc.)
54E35 Metric spaces, metrizability
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] van Douwen, Proc. Amer. Math. Soc 95 pp 101– (1985)
[2] Good, Topology and its Applications 63 pp 79– (1995)
[3] , and , On Stone’s theorem and the axiom of choice. Preprint 1996.
[4] Howard, Math. Logic Quarterly 44 pp 367– (1998)
[5] and , Weak forms of the Axiom of Choice. In Preparation.
[6] The Axiom of Choice. North-Holland Publ. Comp., Amsterdam 1973. · Zbl 0259.02051
[7] Levy, Fund. Math. 50 pp 475– (1962)
[8] Loś, Fund. Math. 41 pp 49– (1954)
[9] Pincus, Annals Math. Logic 11 pp 105– (1977)
[10] Sorgenfrey, Bull. Amer. Math. Soc. 53 pp 631– (1947)
[11] and , Counterexamples in Topology. Springer-Verlag, Berlin-Heidelberg-New York 1986.
[12] General Topology. Addison-Wesley Publ. Co., Reading, Ma, 1968.
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.