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 compactT2 space is compact. TA2: An amorphous power of a compactT2 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 axiomD3 (a set is wellorderable, if each of its infinite subsets is structured) together with RT.
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