On a notion of smallness for subsets of the Baire space
Author:
Alexander S. Kechris
Journal:
Trans. Amer. Math. Soc. 229 (1977), 191207
MSC:
Primary 04A15; Secondary 54H05
MathSciNet review:
0450070
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Abstract: Let us call a set of functions from into bounded if there is a countable sequence of functions such that every member of A is pointwise dominated by an element of that sequence. We study in this paper definability questions concerning this notion of smallness for subsets of . We show that most of the usual definability results about the structure of countable subsets of have corresponding versions which hold about bounded subsets of . For example, we show that every bounded subset of has a ``bound'' and also that for any there are largest bounded and sets. We need here the axiom of projective determinacy if . In order to study the notion of boundedness a simple game is devised which plays here a role similar to that of the standard games (see [My]) in the theory of countable sets. In the last part of the paper a class of games is defined which generalizes the  and  (or BanachMazur) games (see [My]) as well as the game mentioned above. Each of these games defines naturally a notion of smallness for subsets of whose special cases include countability, being of the first category and boundedness and for which one can generalize all the main results of the present paper.
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 J. E. Fenstad, The axiom of determinateness, Proc. Second Scandinavian Logic Sympos., NorthHolland, Amsterdam, pp. 4161. MR 0332479 (48:10806)
 [Fr]
 H. Friedman, Borel sets and hyperdegrees, J. Symbolic Logic 38 (1973), 405409. MR 49 #30. MR 0335248 (49:30)
 [Fr]
 , A basis theorem for L (circulated notes).
 [Gu]
 D. Guaspari, Thin and wellordered analytical sets, Ph.D. Thesis, Univ. of Cambridge, 1972.
 [H]
 S. Hechler, On the existence of certain cofinal subsets of , Proc. Sympos. Pure Math., vol. 13, part 2, Amer. Math. Soc., Providence, R.I., 1974, pp. 155173. MR 50 #12716. MR 0360266 (50:12716)
 [Ke]
 A. S. Kechris, Lecture notes on descriptive set theory, M.I.T., Cambridge, Mass., 1973.
 [Ke]
 , The theory of countable analytical sets, Trans. Amer. Math. Soc. 202 (1975), 259298. MR 0419235 (54:7259)
 [Ke]
 , Measure and category in effective descriptive set theory, Ann. Math. Logic 5 (1972/73), 337384. MR 51 #5308. MR 0369072 (51:5308)
 [Ku]
 K. Kuratowski, Topology, Vol. 1, Academic Press, New York, 1966. MR 36 #840. MR 0217751 (36:840)
 [Ma]
 D. A. Martin, determinacy implies determinacy, 1973 (circulated notes).
 [Ma]
 , Countable sets, 1972 (circulated notes).
 [Mo]
 Y. N. Moschovakis, Descriptive set theory, NorthHolland (to appear). MR 561709 (82e:03002)
 [Mo]
 , Determinacy and prewellorderings of the continuum, Mathematical Logic and Foundations of Set Theory (Proc. Internat. Colloq., Jerusalem, 1968), NorthHolland, Amsterdam, 1970, pp. 2462. MR 43 #6082. MR 0280362 (43:6082)
 [Mo]
 , Analytical definability in a playful universe, Proc. Fourth Internat. Congr. Logic, Methodology and Philosophy of Science, (Bucharest, 1971), NorthHolland, Amsterdam, 1973. MR 0540769 (58:27486)
 [Mo]
 , Uniformization in a playful universe, Bull. Amer. Math. Soc. 77 (1971), 731736. MR 44 #2609. MR 0285390 (44:2609)
 [My]
 J. Mycielski, On the axiom of determinateness, Fund. Math. 53 (1964), 205224. MR 28 #4991. MR 0161787 (28:4991)
 [MyS]
 J. Mycielski and S. Swierczkowski, On the Lebesgue measurability and the axiom of determinateness, Fund. Math. 54 (1964), 6771. MR 28 #4992. MR 0161788 (28:4992)
 [O]
 J. C. Oxtoby, Measure and category, Springer, New York, 1971. MR 584443 (81j:28003)
 [R]
 H. Rogers, Jr., Theory of recursive functions and effective computability, McGrawHill, New York, 1967. MR 37 #61. MR 0224462 (37:61)
 [Sh]
 J. R. Shoenfield, Mathematical logic, AddisonWesley, Reading, Mass., 1967. MR 37 #1224. MR 0225631 (37:1224)
 [St]
 J. Stern, Some measuretheoretic results in effective descriptive set theory, Israel J. Math. 20 (1975), 97110. MR 0387057 (52:7904)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029947197704500701
PII:
S 00029947(1977)04500701
Article copyright:
© Copyright 1977
American Mathematical Society
