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Keywords:
property $(a)$; dominating families; small cardinals; inner models of measurability
Summary:
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.
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