ISSN:
1432-0665
Keywords:
Primary: 03E25, 05A17, 54D30
;
Secondary:54A35, 54B10, 04A25, 03E35
Source:
Springer Online Journal Archives 1860-2000
Topics:
Mathematics
Notes:
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.
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1007/BF02007144
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