Geodesic
Almost disjoint families and “never” cardinal invariants
Commentationes Mathematicae Universitatis Carolinae, Tome 50 (2009) no. 3, pp. 433-444.

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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 never soft and 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., On the presence of countable paracompactness, normality and property $(a)$ in spaces from almost disjoint families, Questions Answers Gen. Topology 25(2007), no. 1, 1--18, and give some information about the strength of this principle.
Classification : 03E17, 03E65, 54A35, 54D20
Keywords: almost disjoint families; parametrized weak diamond principles; property $(a)$; countable paracompactness 
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     title = {Almost disjoint families and {\textquotedblleft}never{\textquotedblright} cardinal invariants},
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Morgan, Charles; da Silva, Samuel Gomes. Almost disjoint families and “never” cardinal invariants. Commentationes Mathematicae Universitatis Carolinae, Tome 50 (2009) no. 3, pp. 433-444. http://geodesic.mathdoc.fr/item/CMUC_2009__50_3_a9/