Baire-Type Properties in Metrizable c0(Ω, X)

Ferrando and Lüdkovsky proved that for a non-empty set Ω normed space X, the c0(Ω,X) is barrelled, ultrabornological, or unordered Baire-like if only X is, respectively, Baire-like. When metrizable locally convex space, with an increasing sequence of semi-norms .n∈N defining its topology, then over field K (of real complex numbers) all functions f:Ω→X such each ε>0 n∈N ω∈Ω:f(ω)n>ε finite empty, topology defined by fn=supf(ω)n:ω∈Ω, n∈N. Kąkol, López-Pellicer Moll-López also quasi bornological, Baire-like, totally barrelled class p p. The main result this paper baireled baireled, proof divided in several lemmas, aim making it easier to read. An application closed graph theorem, two open problems are presented.