Geodesic
Property $(a)$ and dominating families
Commentationes Mathematicae Universitatis Carolinae, Tome 46 (2005) no. 4, pp. 667-684.

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Generalizations of earlier negative results on Property $(a)$ are proved and two questions on an $(a)$-version of Jones' Lemma are posed. We discuss these questions in the realm of locally compact spaces. Using dominating families of functions as a tool, we prove that under the assumptions ``$2^\omega$ is regular'' and ``$2^\omega 2^{\omega_1}$'' the existence of a $T_1$ separable locally compact $(a)$-space with an uncountable closed discrete subset implies the existence of inner models with measurable cardinals. We also use cardinal invariants such as $\frak d$ to prove results in the class of locally compact spaces that strengthen, in such class, the negative results mentioned above.
Classification : 03E04, 54A25, 54A35, 54D20
Keywords: property $(a)$; dominating families; small cardinals; inner models of measurability 
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da Silva, Samuel Gomes. Property $(a)$ and dominating families. Commentationes Mathematicae Universitatis Carolinae, Tome 46 (2005) no. 4, pp. 667-684. http://geodesic.mathdoc.fr/item/CMUC_2005__46_4_a6/