It is shown that in ZF Martin's 
-axiom
together with the axiom of countable choice for finite sets imply that
arbitrary powers
2X of a 2-point discrete space are
Baire; and that the latter property implies the following: (a) the
axiom of countable choice for finite sets, (b) power sets of infinite
sets are Dedekind-infinite, (c) there are no amorphous sets, and (d)
weak forms of the Kinna-Wagner principle.
References
[1] Blass, A., ``Injectivity, projectivity, and the axiom of choice,'' Transactions of the American Mathematical Society, vol. 255 (1979), pp. 31--59.
[2] Brunner, N., ``Sequential compactness and the axiom of choice,'' Notre Dame Journal of Formal Logic, vol. 24 (1983), pp. 89--92.
[3] Fossy, J., and M. Morillon, ``The Baire category property and some notions of compactness,'' Journal of the London Mathematical Society, vol. 57 (1998), pp. 1--19.
[4] Herrlich, H., and K. Keremedis, ``Products, the Baire category theorem, and the axiom of dependent choice,'' Commentationes Mathematicae Universitatis Carolinae, vol. 40 (1999), pp. 771--75.
[5] Howard P., and J. E. Rubin, ``Consequences of the Axiom of Choice,'' American Mathematical Society, Mathematical Surveys and Monographs, vol. 59 (1998).
[6] Łos, J., and C. Ryll-Nardzewski, ``Effectiveness of the representation theory for Boolean algebras,'' Fundamenta Mathematicæ, vol. 42 (1955), pp. 49--56.
[7] Mycielski, J., ``Two remarks on Tychonoff's product theorem,'' Bulletin L'Académie Polonaise des Science, Série des Sciences Mathématiques, Astronomiques et Physiques, vol. 12 (1964), pp. 439--41.
[8] Rubin, H. and D. Scott, ``Some topological theorems equivalent to the Boolean prime ideal theorem,'' Bulletin of the American Mathematical Society, vol. 60 (1954), p. 389.
[9] Shannon, G. P., ``Provable forms of Martin's axiom,'' Notre Dame Journal of Formal Logic, vol. 31 (1990), pp. 382--88.
