On b-boundary in differential spaces category

The aim of this paper is to present a new conceptual approach towards the well-known Schmidt's b-boundary of spacetime. Instead of working in the category of infinitely differentiable manifolds (called also smooth manifolds), we switch to the category of Sikorski differential spaces. Yet, such an approach was developed some time ago, but in this paper, we develop it further. In particular, we mix it with algebraic approach á la Nestruev. It lead us in a natural way towards the new concept of so called generator boundary (in short: gen-boundary). Surprisingly, such a study in the context of cosmology has not been done yet. We provide a through explanation of jointed Nestruev and Sikorski's formalisms. Further, as a result, the concept of gen-boundary emerges. We discuss its properties. Afterwards, having the developed techniques at hand, we re-discuss the Schmidt's b-boundary in this new (more algebraic) language. Interestingly, a classification of spacetime singularities naturally emerges from the proposed formalism.