Sober approach spaces

摘要

In this article we will study a notion of sobriety for approach spaces, modelled after the corresponding familiar concept in the case of topological spaces. Recall that the latter is closely linked to the relationship between the categories Top of topological spaces and Frm of frames given by a pair of contravariant functors which assign to each topological space its frame of open subsets and to each frame its spectrum. Sober spaces are those spaces for which the corresponding adjunction map is a homeomorphism. Frames appear very naturally in the setting of approach spaces, since the so-called regular function frame of an approach space, which takes the role of the frame of open sets in topology, actually is a frame in the strict definition. It is however equipped with supplementary structure and accordingly we need to introduce an abstractly defined version, called an approach frame. This setting too will lead to a contravariant functor R from the category AP of approach spaces to the category AFrm of approach frames and a spectrum functor Σ in the opposite direction. An approach space X is then called sober if the corresponding adjunction map X → ΣRX is an isomorphism. Of course we give an account of the basic facts concerning the functors R and Σ and present a number of results about sobriety and spatiality. However, we also prove a surprising relation between sobriety and completeness and between sobrification and completion for uniform approach spaces. These results bring to the foreground a completeness-aspect of the notion of sobriety which is somewhat hidden in the topological setting.