A Note on the Topologicity of Quantale-Valued Topological Spaces

For a quantale V, the category V-Top of V-valued topological spaces may be introduced as a full subcategory of those V-valued closure spaces whose closure operation preserves finite joins. In generalization of Barr’s characterization of topological spaces as the lax algebras of a lax extension of the ultrafilter monad from maps to relations of sets, for V completely distributive, V-topological spaces have recently been shown to be characterizable by a lax extension of the ultrafilter monad to V-valued relations. As a consequence, V-Top is seen to be a topological category over Set, provided that V is completely distributive. In this paper we give a choice-free proof that V-Top is a topological category over Set under the considerably milder provision that V be a spatial coframe. When V is a continuous lattice, that provision yields complete distributivity of V in the constructive sense, hence also in the ordinary sense whenever the axiom of choice is granted.