Weight of precompact subsets and tightness ✩

Pfister (1976) and Cascales and Orihuela (1986) proved that precompact sets in (DF)- and (LM)-spaces have countable weight, i.e., are metrizable. Improvements by Valdivia (1982), Cascales and Orihuela (1987), and K¸akol and Saxon (preprint) have varying methods of proof. For these and other improvements a refined method of upper semicontinuous compact-valued maps applied to uniform spaces will suffice. At the same time, this method allows us to dramatically improve Kaplansky’s theorem, that the weak topology of metrizable spaces has countable tightness, extending it to include all (LM)- spaces and all quasi-barrelled (DF)-spaces, both in the weak and original topologies. One key is showing that for a large class G including all (DF)- and (LM)-spaces, countable tightness of the weak topology of E in G is equivalent to realcompactness of the weak∗ topology of the dual of E.  2002 Elsevier Science (USA). All rights reserved.