## Samuel Gomes da Silva

*Property $(a)$ and dominating families *

Comment.Math.Univ.Carolinae 46,4 (2005) 667-684. **Abstract:**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.

**Keywords:** property $(a)$, dominating families, small cardinals, inner models of measurability

**AMS Subject Classification:** Primary 54A25, 54D20; Secondary 54A35

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