Borel games and the Baire property

Authors:
Kenneth Schilling and Robert Vaught

Journal:
Trans. Amer. Math. Soc. **279** (1983), 411-428

MSC:
Primary 04A15; Secondary 03C15, 54H05

DOI:
https://doi.org/10.1090/S0002-9947-1983-0704624-8

MathSciNet review:
704624

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: The Borel game operations are a natural generalization of the operation A. It is shown that these operations preserve the property of Baire in all topological spaces. Applications are given to invariant descriptive set theory and the model theory of infinitary logic.

**[1]**J. Burgess,*Infinitary languages and descriptive set theory*, Ph.D. thesis, Univ. of California, Berkeley, Calif., 1974.**[2]**J. E. Fenstad and D. Normann,*On absolutely measurable sets*, Fund. Math.**81**(1974), 91-98. MR**0338299 (49:3065)****[3]**D. Gale and F. Stewart,*Infinite games with perfect information*. Contributions to the Theory of Games, Vol. II, Ann. of Math. Studies, no. 28, Princeton Univ. Press, Princeton, N. J., 1953, pp. 254-266. MR**0054922 (14:999b)****[4]**A. Kechris,*Measure and category in effective descriptive set theory*, Ann. Math. Logic**5**(1973), 337-384. MR**0369072 (51:5308)****[5]**-,*Forcing in analysis*, Higher Set Theory (Proc. Conf. Oberwolfach, 1977), Lecture Notes in Math., vol. 669, Springer, Berlin, 1978, pp. 277-302. MR**520191 (80c:03051)****[6]**K. Kuratowski,*Topology*, Vol. I, Academic Press, New York; PWN, Warsaw, 1966. MR**0217751 (36:840)****[7]**D. A. Martin,*Borel determinacy*, Ann. of Math. (2)**102**(1975), 363-371. MR**0403976 (53:7785)****[8]**D. Miller,*Invariant descriptive set theory and the topological approach to model theory*, Ph.D. thesis, Univ. of California, Berkeley, Calif., 1976.**[9]**J. C. Oxtoby,*The Banach-Mazur game and Banach category theorem*. Contributions to the Theory of Games, Vol. III, Ann. of Math. Studies, no. 28, Princeton Univ. Press, Princeton, N. J., 1957, pp. 159-163. MR**0093741 (20:264)****[10]**G. Reyes,*Typical and generic relations in a Baire space for models*, Ph.D. thesis, Univ. of California, Berkeley, Calif., 1967.**[11]**K. Schilling,*Absolutely**operations and the Baire property*, Abstracts Amer. Math. Soc.**1**(1980), 239. Abstract #80T-E22.**[12]**R. Sikorski,*Boolean algebras*, 3rd ed., Ergeb. Math. Grenzgeb., Bd. 25, Springer, New York, 1969. MR**0242724 (39:4053)****[13]**R. Vaught,*Invariant sets in topology and logic*, Fund. Math.**82**(1974), 269-294. MR**0363912 (51:167)****[14]**R. Vaught and K. Schilling,*Borel game operations and the Baire property*, Notices Amer. Math. Soc.**26**(1979), A-247. Abstract #764-E1.

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC:
04A15,
03C15,
54H05

Retrieve articles in all journals with MSC: 04A15, 03C15, 54H05

Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1983-0704624-8

Article copyright:
© Copyright 1983
American Mathematical Society