Property ( a ) and dominating families Samuel

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 is regular” and “2 < 21” the existence of a T1 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 d to prove results in the class of locally compact spaces that strengthen, in such class, the negative results mentioned above.