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Almost Disjoint Families and Diagonalizations of Length Continuum

Published online by Cambridge University Press:  15 January 2014

Dilip Raghavan
Dilip Raghavan*
Affiliation:
Department of Mathematics, University of Toronto, Toronto, ON M5S 2E4, Canada. E-mail:raghavan@math.toronto.edu, URL: http://www.math.toronto.edu/raghavan
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Abstract

We present a survey of some results and problems concerning constructions which require a diagonalization of length continuum to be carried out, particularly constructions of almost disjoint families of various sorts. We emphasize the role of cardinal invariants of the continuum and their combinatorial characterizations in such constructions.

Type
Research Article
Information
Bulletin of Symbolic Logic , Volume 16 , Issue 2 , June 2010 , pp. 240 - 260
Copyright
Copyright © Association for Symbolic Logic 2010

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References

REFERENCES

[1] Balcar, B., Dočkálková, J., and Simon, P., Almost disjoint families of countable sets, Finite and infinite sets, vol. I, II (Eger, 1981), Colloquia Mathematics Societas János Bolyai, vol. 37, North-Holland, Amsterdam, 1984, pp. 59–88.Google Scholar
[2] Balcar, B., Pelant, J., and Simon, P., The space of ultrafilters on N covered by nowhere dense sets, Polska Akademia Nauk. Fundamenta Mathematicae, vol. 110 (1980), no. 1, pp. 11–24.Google Scholar
[3] Balcar, B. and Simon, P., Disjoint refinement, Handbook of Boolean algebras, vol. 2, North-Holland, Amsterdam, 1989, pp. 333–388.Google Scholar
[4] Balcar, B. and Vojtáš, P., Almost disjoint refinement of families of subsets of N, Proceedings of the American Mathematical Society, vol. 79 (1980), no. 3, pp. 465–470.Google Scholar
[5] Bartoszyński, T., Combinatorial aspects of measure and category, Polska Akademia Nauk. Fundamenta Mathematicae, vol. 127 (1987), no. 3, pp. 225–239.Google Scholar
[6] Bartoszyński, T., Invariants of measure and category, Handbook of set theory, to appear.Google Scholar
[7] Bartoszyński, T. and Judah, H., Set theory: On the structure of the real line, A K Peters Ltd., Wellesley, MA, 1995.Google Scholar
[8] Baumgartner, J. E., Chains and antichains in P(ω), The Journal of Symbolic Logic, vol. 45 (1980), no. 1, pp. 85–92.Google Scholar
[9] Blass, A., Combinatorial cardinal characteristics of the continuum, Handbook of set theory, to appear.Google Scholar
[10] Brendle, J., MOB families and MAD families , Archive for Mathematical Logic, vol. 37 (1997), no. 3, pp. 183–197.Google Scholar
[11] Brendle, J. and Hrušák, M., Countable Fréchet Boolean groups: An independence result, The Journal of Symbolic Logic, (to appear).Google Scholar
[12] Brendle, J. and Yatabe, S., Forcing indestructibility of MAD families, Annals of Pure and Applied Logic, vol. 132 (2005), no. 2-3, pp. 271–312 Google Scholar
[13] Farah, I., A proof of the -absoluteness theorem, Advances in logic, Contemporary Mathematicians, vol. 425, American Mathematical Society, Providence, RI, 2007, pp. 9–22.Google Scholar
[14] Fuchino, S., Koppelberg, S., and Shelah, S., Partial orderings with the weak Freese–Nation property, Annals of Pure and Applied Logic, vol. 80 (1996), no. 1, pp. 35–54.Google Scholar
[15] García-Ferreira, S., Continuous functions between Isbell–Mrówka spaces, Commentationes Mathematicae Universitatis Carolinae, vol. 39 (1998), no. 1, pp. 185–195.Google Scholar
[16] Gruenhage, G. and Szeptycki, P. J., Fréchet–Urysohn for finite sets, Topology and its Applications, vol. 151 (2005), no. 1-3, pp. 238–259.Google Scholar
[17] Hindman, N., On the existence of c-points in βN/N, Proceedings of the American Mathematical Society, vol. 21 (1969), pp. 277–280.Google Scholar
[18] Hrušák, M., MAD families and the rationals, Commentationes Mathematicae Universitatis Carolinae, vol. 42 (2001), no. 2, pp. 345–352.Google Scholar
[19] Hrušák, M. and Ferreira, S. García, Ordering MAD families a la Katétov, The Journal of Symbolic Logic, vol. 68 (2003), no. 4, pp. 1337–1353.Google Scholar
[20] Kastermans, B., Very MAD families, Advances in logic, Contemporary Mathematicians, vol. 425, American Mathematical Society, Providence, RI, 2007, pp. 105–112.Google Scholar
[21] Larson, P. B., Almost-disjoint coding and strongly saturated ideals, Proceedings of the American Mathematical Society, vol. 133 (2005), no. 9, pp. 2737–2739.Google Scholar
[22] Leathrum, T. E., A special class of almost disjoint families, The Journal of Symbolic Logic, vol. 60 (1995), no. 3, pp. 879–891.Google Scholar
[23] Miller, A.W., Arnie Miller's problem list, Set theory of the reals ( Ramat Gan , 1991), Israel Mathematical Conference Proceedings, vol. 6, Bar-Ilan University, Ramat Gan, 1993, pp. 645–654.Google Scholar
[24] Moore, J. T., A solution to the L space problem, Journal of the American Mathematical Society, vol. 19 (2006), no. 3, pp. 717–736, (electronic).Google Scholar
[25] Pierce, R. S., Modules over commutative regular rings, Memoirs of the American Mathematical Society, no. 70, American Mathematical Society, Providence, R.I., 1967.Google Scholar
[26] Raghavan, D., Maximal almost disjoint families of functions, Fundamenta Mathematicae, vol. 204 (2009), no. 3, pp. 241–282.Google Scholar
[27] Raghavan, D., A model with no strongly separable almost disjoint families, to appear.Google Scholar
[28] Raghavan, D., There is a Van Douwen MAD family, Transactions of the American Mathematical Society, to appear.Google Scholar
[29] Raghavan, D. and Stepräns, J., On weakly tight families, to appear.Google Scholar
[30] Shelah, S., MAD families and SANE player, preprint, 0904.0816.Google Scholar
[31] Shelah, S. and Stepräns, J., MASAs in the Calkin algebra without the continuum hypothesis, Canadian Mathematical Bulletin.Google Scholar
[32] Szentmiklóssy, Z., S-spaces and L-spaces under Martin's axiom, Topology, Vol. II ( Proc. Fourth Colloq. , Budapest, 1978), Colloquia Mathematics Societas János Bolyai, vol. 23, North-Holland, Amsterdam, 1980, pp. 1139–1145.Google Scholar
[33] Todorčević, S., Partition problems in topology, Contemporary Mathematics, vol. 84, American Mathematical Society, Providence, RI, 1989.Google Scholar
[34] Todorčević, S., A dichotomy for P-ideals of countable sets, Fundamenta Mathematicae, vol. 166 (2000), no. 3, pp. 251–267.Google Scholar
[35] Todorčević, S., Walks on ordinals and their characteristics, Progress in Mathematics, vol. 263, Birkhäuser Verlag, Basel, 2007.Google Scholar
[36] van Douwen, E. K. and Kunen, K., L-spaces and S-spaces in P(ω), Topology and its Applications, vol. 14 (1982), no. 2, pp. 143–149.Google Scholar
[37] Zhang, Y., Towards a problem of E. van Douwen and A. Miller, Mathematical Logic Quarterly, vol. 45 (1999), no. 2, pp. 183–188.Google Scholar