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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
 MSC Classifications: 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