Combinatorial aspects of F_{𝜎} filters with an application to 𝒩-sets

Laflamme, Claude

doi:10.1090/S0002-9939-97-03926-9

Combinatorial aspects of F$_\sigma$ filters with an application to $\mathcal N$-sets
HTML articles powered by AMS MathViewer

by Claude Laflamme PDF

Proc. Amer. Math. Soc. 125 (1997), 3019-3025 Request permission

Abstract:

We discuss F$_\sigma$ filters and show that the minimum size of a filter base generating an undiagonalizable filter included in some F$_\sigma$ filter is the better known bounded evasion number ${\frak e}_{ubd}$. An application to $\mathcal {N}$-sets from trigonometric series is given by showing that if $A$ is an $\mathcal {N}$-set and $B$ has size less than ${\frak e}_{ubd}$, then $A \cup B$ is again an $\mathcal {N}$-set.

References

Tomek Bartoszyński, Combinatorial aspects of measure and category, Fund. Math. 127 (1987), no. 3, 225–239. MR 917147, DOI 10.4064/fm-127-3-225-239
Tomek Bartoszyński and Marion Scheepers, Remarks on sets related to trigonometric series, Topology Appl. 64 (1995), no. 2, 133–140. MR 1340865, DOI 10.1016/0166-8641(94)00077-G
N. K. Bary, A treatise on trigonometric series. Vols. I, II, A Pergamon Press Book, The Macmillan Company, New York, 1964. Authorized translation by Margaret F. Mullins. MR 0171116
Andreas Blass, Cardinal characteristics and the product of countably many infinite cyclic groups, J. Algebra 169 (1994), no. 2, 512–540. MR 1297160, DOI 10.1006/jabr.1994.1295
Jörg Brendle, Evasion and prediction—the Specker phenomenon and Gross spaces, Forum Math. 7 (1995), no. 5, 513–541. MR 1346879, DOI 10.1515/form.1995.7.513
J. Brendle and S. Shelah, Evasion and Prediction II, J. London Math. Soc. (2) 53 (1996), no. 1, 19-27.
Z. Bukovská and L. Bukovský, Adding Small Sets to an N-set, Proc. Amer. Math. Soc. (to appear, 1994).
Maryvonne Daguenet, Emploi des filtres sur $N$ dans l’étude descriptive des fonctions, Fund. Math. 95 (1977), no. 1, 11–33 (French). MR 470928, DOI 10.4064/fm-95-1-11-33
Claude Laflamme, Zapping small filters, Proc. Amer. Math. Soc. 114 (1992), no. 2, 535–544. MR 1068126, DOI 10.1090/S0002-9939-1992-1068126-5
Saharon Shelah, Proper forcing, Lecture Notes in Mathematics, vol. 940, Springer-Verlag, Berlin-New York, 1982. MR 675955
Saharon Shelah, Vive la différence. I. Nonisomorphism of ultrapowers of countable models, Set theory of the continuum (Berkeley, CA, 1989) Math. Sci. Res. Inst. Publ., vol. 26, Springer, New York, 1992, pp. 357–405. MR 1233826, DOI 10.1007/978-1-4613-9754-0_{2}0
J. Vaughan, Small Uncountable Cardinals and Topology, Open problems in Topology (J. van Mill and G. Reed, eds.), North-Holland, 1990, 195-218.