On ultraregular inductive limits

An inductive limit (E, t) = ind(E-n, t(n)) is said to have property (P) if every closed absolutely convex neighborhood in (E-n, t(n)) is closed in (En+1, t(n+1)). This property was introduced and investigated by J. Kucera. In this paper we give some equivalent descriptions of property (P) and prove that property (P) implies ultraregularity. Particularly, if all (E-n, t(n)) are metrizable locally convex spaces, we have: (E, t) is ultraregular if and only if (E, t) is a strict inductive limit and for each n is an element of N, there is m = m(n) is an element of N such that (E) over bar (E)(n) subset of E-m; (E, t) has property (P) if and only if (E, t) is a strict inductive limit and each E-n is closed in (En+1, t(n+1)).