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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
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