Metrizable (LF)-spaces, (db)-spaces, and the separable quotient problem

The existence of metrizable (LF)-spaces was announced by Stephen A. Saxon (“Metrizable generalized (LF)-spaces”, 701–46–14), in Notices Amer. Math. Soc. 20 (1973), A–143. Elsewhere, the authors have discovered an abundant existence of metrizable and normable (generalized) (LF)-spaces, while observing that an (LF)-space is metrizable if and only if it is Baire-like. Recently, W. Robertson, I. Tweddle and F.E. Yeomans introduced the class of locally convex spaces E having the property(db) if E is the union of an increasing sequence (En) of vector subspaces, then some En is dense and barrelled.