α-Dedekind complete archimedean vector lattices versus α-quasi- F spaces

Abstract

α denotes an uncountable cardinal number, and W (α) denotes the category of archimedean vector lattices with distinguished waek unit and unit preserving vector lattice homomorphisms. In this paper we show that in W (α), the full subcategory of α-Dedekind complete objects is epireflective. We explain how the Yosida functor connects the algebraic notions of α-Dedekind complete, α-dense, and α-jam-dense with the topological notions of α-quasi- F, α-irreducible, and α-SpFi morphism. We then go on to show that the α-quasi- F cover of a compact space X, denoted (QF α X, q α) is the Yosida space of the α-Dedekind complete epireflection of C( X) in W (α). Finally we show that in the topological category α-SpFi, the full subcategory of α-quasi- F spaces is monocoreflective, and for each compact space X,(QF α X, q α) is the α-quasi- F monocoreflection of X.