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On Weakly Tight Families

  Published:2012-08-25
 Printed: Dec 2012
  • Dilip Raghavan,
    Department of Mathematics, University of Toronto, Toronto, ON M5S 2E4
  • Juris Steprāns,
    Department of Mathematics, York University, Toronto, ON M3J 1P3
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Abstract

Using ideas from Shelah's recent proof that a completely separable maximal almost disjoint family exists when $\mathfrak{c} \lt {\aleph}_{\omega}$, we construct a weakly tight family under the hypothesis $\mathfrak{s} \leq \mathfrak{b} \lt {\aleph}_{\omega}$. The case when $\mathfrak{s} \lt \mathfrak{b}$ is handled in $\mathrm{ZFC}$ and does not require $\mathfrak{b} \lt {\aleph}_{\omega}$, while an additional PCF type hypothesis, which holds when $\mathfrak{b} \lt {\aleph}_{\omega}$ is used to treat the case $\mathfrak{s} = \mathfrak{b}$. The notion of a weakly tight family is a natural weakening of the well studied notion of a Cohen indestructible maximal almost disjoint family. It was introduced by Hrušák and García Ferreira, who applied it to the Katétov order on almost disjoint families.
Keywords: maximal almost disjoint family, cardinal invariants maximal almost disjoint family, cardinal invariants
MSC Classifications: 03E17, 03E15, 03E35, 03E40, 03E05, 03E50, 03E65 show english descriptions Cardinal characteristics of the continuum
Descriptive set theory [See also 28A05, 54H05]
Consistency and independence results
Other aspects of forcing and Boolean-valued models
Other combinatorial set theory
Continuum hypothesis and Martin's axiom [See also 03E57]
Other hypotheses and axioms
03E17 - Cardinal characteristics of the continuum
03E15 - Descriptive set theory [See also 28A05, 54H05]
03E35 - Consistency and independence results
03E40 - Other aspects of forcing and Boolean-valued models
03E05 - Other combinatorial set theory
03E50 - Continuum hypothesis and Martin's axiom [See also 03E57]
03E65 - Other hypotheses and axioms
 

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