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Madness and weak forms of normality

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Abstract

We consider weakenings of normality in \(\Psi\)-spaces and prove that the existence of an AD family whose \(\Psi\)-space is almost-normal but not normal follows from CH. In contrast, we prove that it is consistent that no MAD family is almost-normal. We also construct a partly-normal not quasi-normal AD family, answering questions of García-Balan and Szeptycki. We finish by showing that the concepts of almost-normal and strongly \(\aleph_0\)-separated AD families are different, even under CH, answering a question of Oliveira-Rodrigues and Santos-Ronchim.

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References

  1. I. Alshammari, L. Kalantan, and S. A. Thabit, Partial normality, J. Math. Anal., 10 (2019), 1-8.

  2. I. E. AlShammari and L. Kalantan, Quasi-normality of Mrówka spaces, Eur. J. Pure Appl. Math.,13 (2020), 697-700.

  3. B. Balcar, J. Dočkálková, and P. Simon, Almost disjoint families of countable sets, in: Finite and Infinite Sets, Elsevier (1984), 59-88.

  4. B. Balcar and P. Simon, Disjoint refinement, Handbook of Boolean Algebras, vol. 2, North-Holland (Amsterdam 1989), pp. 333-388.

  5. A. Blass, Combinatorial cardinal characteristics of the continuum, in: Foreman M., Kanamori A. (eds), Handbook of Set Theory, Springer (2010), pp. 395-490.

  6. J. Brendle, Dow's principle and Q-sets, Canad. Math. Bull., 42 (1999), 13-24.

  7. A. Dow, On compact separable radial spaces, Canad. Math. Bull., 40 (1997), 422- 432.

  8. A. Dow, Sequential order under PFA, Canad. Math. Bull., 54 (2011), 270-276.

  9. A. Dow and S. Shelah, Martin's axiom and separated mad families, Rend. Circ. Mat. Palermo, 61 (2012), 107-115.

  10. R. Engelking, General Topology, Heldermann (1989).

  11. P. Erdős and S. Shelah, Separability properties of almost-disjoint families of sets, Israel J. Math., 12 (1972), 207-214.

  12. F. Galvin and P. Simon, A Čech function in ZFC, Fund. Math., 193 (2007), 181-188.

  13. O. Guzmán-González, M. Hrušák, C. A. Martínez-Ranero, and U. A. Ramos-García, Generic existence of mad families, J. Symbolic Logic, 82 (2017), 303-316.

  14. S. H. Hechler, Classifying almost-disjoint families with applications to ßN-N, Israel J. Math., 10 (1971), 413-432.

  15. M. Hrušák, Almost disjoint families and topology, in: Recent Progress in General Topology, III, Springer (2014), 601-638.

  16. M. Hrušák and O. Guzmán, n-Luzin gaps, Topology Appl., 160 (2013), 1364-1374.

  17. V. Kannan and M. Rajagopalan, Hereditarily locally compact separable spaces, in: Categorical Topology, Springer (1979), pp. 185-195.

  18. K. Kunen, Set Theory an Introduction to Independence Proofs, Elsevier (2014).

  19. N. N. Luzin, On subsets of the series of natural numbers, Izv. Ross. Akad. Nauk, Ser. Matem., 11 (1947), 403-410.

  20. D. A. Martin and R. M. Solovay, Internal Cohen extensions, Ann. Math. Logic, 2 (1970), 143-178.

  21. H. Mildenberger, D. Raghavan, and J. Steprans, Splitting families and complete separability, Canad. Math. Bull., 57 (2014), 119-124.

  22. S. Mrówka, On completely regular spaces,Fund. Math., 41 (1955), 105-106.

  23. V. dos Oliveira Rodrigues and V. dos Santos Ronchim, Almost-normality of Isbell- Mrówka spaces, Topology Appl., 288 (2020), Paper No. 107470, 13 pp.

  24. D. Raghavan, Almost disjoint families and diagonalizations of length continuum, Bulletin Symbolic Logic, 16 (2010), 240-260.

  25. S. Shelah, MAD saturated families and SANE player,Canad. J. Math., 63 (2011), 1416-1435.

  26. P. Simon, A note on almost disjoint refinement, Acta Univ. Carolin. Math. Phys., 37 (1996), 89-99.

  27. M. Singal and S. P. Arya, Almost-normal and almost completely regular spaces, Glasnik Mat., 5 (1970), 141-152.

  28. P. J. Szeptycki and S. García-Balan, Weak normality properties in \(\Psi\)-spaces, arXiv:2007.05844 (2020).

  29. F. D. Tall, Set-theoretic consistency results and topological theorems concerning the normal Moore space conjecture and related problems, PhD thesis, The University of Wisconsin-Madison (1969).

  30. V. I. Zaitsev, Some classes of topological spaces and their bicompact extensions, Dokl. Akad. Nauk SSSR, 178 (1968), 778-779.

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  1. Centro de Ciencias Matemáticas, Universidad Nacional Autónoma de México, Campus Morelia, 58089, Morelia, Michoacán, México

    C. Corral

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  1. C. Corral
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Correspondence to C. Corral.

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The author gratefully acknowledges support from CONACyT scholarship 742627.

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Corral, C. Madness and weak forms of normality. Acta Math. Hungar. 165, 291–307 (2021). https://doi.org/10.1007/s10474-021-01186-y

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  • DOI: https://doi.org/10.1007/s10474-021-01186-y

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