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Topological properties of incomparable families

Volume 156 / 2019

G. Campero-Arena, J. Cancino, M. Hrušák, D. Meza-Alcántara, F. E. Miranda-Perea Colloquium Mathematicum 156 (2019), 313-323 MSC: 03E17, 03E15, 03E35, 03G05. DOI: 10.4064/cm7403-8-2018 Published online: 15 February 2019

Abstract

We study topological properties of families of mutually incomparable subsets of $\omega $. We say that two subsets $a$ and $b$ of $\omega $ are incomparable if both $a\setminus b$ and $b\setminus a$ are infinite. We raise the question whether there may be an analytic maximal incomparable family and show that (1) it cannot be $K_\sigma $, and (2) every incomparable family with the Baire property is meager. On the other hand, we show that non-meager incomparable families exist in $\mathsf{ZFC}$, while the existence of a non-null incomparable family is consistent. Finally, we show that there are maximal incomparable families which are both meager and null assuming either $\mathfrak r=\mathfrak c$ or the existence of a completely separable MAD family; in particular they exist if $\mathfrak c \lt \aleph _\omega $. Assuming {\sf CH}, we can even construct a maximal incomparable family which is concentrated on a countable set, and hence of strong measure zero.

Authors

  • G. Campero-ArenaDepartamento de Matemáticas
    Facultad de Ciencias
    Universidad Nacional Autónoma de México
    Circuito Exterior S/N
    Ciudad Universitaria, CDMX, 04510 México
    e-mail
  • J. CancinoCentro de Ciencias Matemáticas
    Universidad Nacional Autónoma de México
    Morelia, 58089 México
    e-mail
  • M. HrušákCentro de Ciencias Matemáticas
    Universidad Nacional Autónoma de México
    Morelia, 58089 México
    e-mail
  • D. Meza-AlcántaraDepartamento de Matemáticas
    Facultad de Ciencias
    Universidad Nacional Autónoma de México
    Circuito Exterior S/N
    Ciudad Universitaria, CDMX, 04510 México
    e-mail
  • F. E. Miranda-PereaCentro de Ciencias Matemáticas
    Universidad Nacional Autónoma de México
    Morelia, 58089 México
    e-mail

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