Covering properties which, under weak diamond principles, constrain the extents of separable spaces

Abstract

We show that separable, locally compact spaces with property (a) necessarily have countable extent { i.e., have no uncountable closed, discrete sub- spaces { if the e®ective weak diamond principle }(!; !;<) holds. If the stronger,non-e®ective, diamond principle ©(!; !;<) holds then separable, countably para-compact spaces also have countable extent. We also give a short proof that the latter principle implies there are no small dominating families in !1!.