L-topological spaces as spaces of points

The main background of this paper is a functor, from the category of L-topological spaces to the category of topological spaces, which is defined by means of an attachment (a notion recently introduced) in the lattice L and has very nice properties under the assumption of spatiality. The Pu-Liu's quasi-coincidence relation, largely used to study [0,1]-topological spaces, is determined by a suitable attachment in [0,1], so the results described here apply to that situation. This functor allows to import topological concepts into the fuzzy setting directly from the classical context. Meanwhile, this provides an evaluation of the relevance and of the consistency of (basic) notions in fuzzy topology, as well as a critical view on how these have been introduced and treated up to now. An overview of the relationship between (spatial) attachments and textures is outlined in the last part of the paper.