Every locale (frame) is a dense sublocale of a spatial subfit one

This paper answers a long standing open problem regarding which pointfree topologies (locales) can, or cannot, be embedded into subfit locales. Subfitness, a point-free weak separation property closely related to T1 , is well-known to be non-hereditary. So, the problem amounts to characterizing just how badly non-hereditary the property is.The main result of the paper shows that it is maximally so – every locale can be embedded densely in a subfit locale. Moreover, the locale into which a given locale is embedded, in short the “ambient” locale, can be chosen to be spatial. Among other things, this illustrates the gap between subfitness and T1 for spaces – since any embedding preserves specialization order of points, and T1 spaces have discrete order, there are many (spatial) locales that cannot be embedded in T1 spaces, but all locales can be embedded densely in subfit ones. The main result also highlights an interesting role for points in point-free topology.