For a quantale ${ \sf{ V}}$, the category $\sf V$-${ \bf Top}$ of ${ \sf{ V}}$-valued topological spaces may be introduced as a full subcategory of those ${ \sf{ V}}$-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 ${ \sf{ V}}$ completely distributive, ${ \sf{ V}}$-topological spaces have recently been shown to be characterizable by a lax extension of the ultrafilter monad to ${ \sf{ V}}$-valued relations. As a consequence, ${ \sf{ V}}$-$\bf Top$ is seen to be a topological category over $\bf Set$, provided that ${ \sf{ V}}$ is completely distributive. In this paper we give a choice-free proof that ${ \sf{ V}}$-$\bf Top$ is a topological category over $\bf Set$ under the considerably milder provision that ${ \sf{ V}}$ be a spatial coframe. When ${ \sf{ V}}$ is a continuous lattice, that provision yields complete distributivity of ${ \sf{ V}}$ in the constructive sense, hence also in the ordinary sense whenever the Axiom of Choice is granted.
Categories: cs.LO