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On Weakly Tight Families

Published online by Cambridge University Press:  20 November 2018

Dilip Raghavan  and
Juris Steprāns
Dilip Raghavan
Affiliation:
Department of Mathematics, University of Toronto, Toronto, ON M5S 2E4 email: raghavan@math.toronto.edu
Juris Steprāns
Affiliation:
Department of Mathematics, York University, Toronto, ON M3J 1P3 email: steprans@yorku.ca
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Abstract

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Using ideas from Shelah's recent proof that a completely separable maximal almost disjoint family exists when $c<{{\aleph }_{\omega }}$, we construct a weakly tight family under the hypothesis $\mathfrak{s}\le \mathfrak{b}<{{\aleph }_{\omega }}$. The case when $\mathfrak{s}<\mathfrak{b}$ is handled in ZFC and does not require $\mathfrak{b}<{{\aleph }_{\omega }}$, while an additional PCF type hypothesis, which holds when $\mathfrak{b}<{{\aleph }_{\omega }}$ 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 [8], who applied it to the Katétov order on almost disjoint families.

Keywords

03E1703E1503E3503E4003E0503E5003E65maximal almost disjoint familycardinal invariants
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 64 , Issue 6 , 01 December 2012 , pp. 1378 - 1394
Copyright
Copyright © Canadian Mathematical Society 2012

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