Charles Morgan, Samuel Gomes da Silva
Almost disjoint families and ``never'' cardinal invariants

Comment.Math.Univ.Carolinae 50,3 (2009) 433-444.

Abstract:We define two cardinal invariants of the continuum which arise naturally from combinatorially and topologically appealing properties of almost disjoint families of sets of the natural numbers. These are the {\it never soft\/} and {\it never countably paracompact\/} numbers. We show that these cardinals must both be equal to $\omega_1$ under the effective weak diamond principle $\diamondsuit (\omega,\omega,<)$, answering questions of da Silva S.G., {\it On the presence of countable paracompactness, normality and property $(a)$ in spaces from almost disjoint families\/}, Questions Answers Gen. Topology {\bf 25} (2007), no.~ 1, 1--18, and give some information about the strength of this principle.

Keywords: almost disjoint families, parametrized weak diamond principles, property $(a)$, countable paracompactness
AMS Subject Classification: 03E65 54D20 03E17 54A35

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