The Čech-Stone compactification of the reals and coz-included sublocales

A new notion of a coz-included sublocale is introduced. It is a necessary condition for a sublocale to be a dense C-embedding, and together with the property of being well-embedded, is sufficient. However, although it is clearly a strengthening of coz-ontoness (or z-embeddings), it is incomparable with C^*-embeddings as is demonstrated by the Čech-Stone compactification of the real numbers. Motivated by non-trivial spatial examples, this new notion and other weaker variants, such as casi coz-included and relative zero sublocales, are explored and characterized in the point-free setting. As a consequence, new characterizations of several classes of frames, such as perfectly normal and $Oz$ frames, among others, are provided. Besides the lattice theoretic interpretations, there are some interesting topological ramifications.