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Amorphe Potenzen kompakter Räume

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Abstract

A set is amorphous, if it is not a union of two disjoint infinite subsets. The following variants of the Tychonoff product theorem are investigated in the hierarchy of weak choice principles. TA1: An amorphous power of a compactT 2 space is compact. TA2: An amorphous power of a compactT 2 space which as a set is wellorderable is compact. In ZF0TA1 is equivalent to the assertion, that amorphous sets are finite. RT is Ramsey's theorem, that every finite colouring of the set ofn-element subsets of an infinite set has an infinite homogeneous subset and PW is Rubin's axiom, that the power set of an ordinal is wellorderable. In ZF0RT+PW implies TA2. Since RT+PW is compatible with the existence of infinite amorphous sets, TA2 does not imply TA1 in ZF0. But TA2 cannot be proved in ZF0 alone. As an application, we prove a theorem of Stone, using a weak wellordering axiomD 3 (a set is wellorderable, if each of its infinite subsets is structured) together with RT.

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  1. Universität für Bodenkultur-Mathematik, Gregor-Mendel-Straße 33, A-1180, Wien, Österreich

    Norbert Brunner

  2. Purdue University — Mathematics, 47907, West Lafayette, IN, USA

    Norbert Brunner

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Diese Arbeit ist Teil der Habilitationsschrift des Verfassers im Fachgebiet „Mathematische Analysis“ an der Technischen Universität Wien.

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Brunner, N. Amorphe Potenzen kompakter Räume. Arch math Logik 24, 119–135 (1984). https://doi.org/10.1007/BF02007144

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