A note on closed discrete subsets of separable (a)-spaces

We show that the existence of a T1 separable space with an uncountable closed discrete subset which satisfies relative versions of property (a) and local compactness implies the existence of small dominating families in the family of functions of ?1 into ?. Considering well-known relationships between small dominating families and large cardinals, it follows that if Y is an uncountable closed discrete subset of a T1 separable (a)-space X then there is no way to prove within ZFC that Y satisfies relative local compactness.