Extensions of the ring of continuous functions generated by regular, countably-divisible, complete rings of quotients, and their corresponding pre-images

In this article we consider metaregular and countably-divisible extensions generated by a regular quotient ring of the ring of continuous functions in the spirit of Fine–Gillman–Lambek. The corresponding pre-images of maximal ideals are considered in connection with these extensions. These pre-images are called small absolutes and a-nonconnected coverings. To characterize these structures a new topological structure is introduced for Aleksandrov spaces with a precovering. In this connection we introduce the notion of a non-connected covering of step type. In the first part of the article we give a characterization of a small absolute as a relatively countably non-connected covering (Theorem 1). We also give a description of the absolute (Theorem 2) and of Aleksandrov pre-images of maximal ideals of Hausdorff–Sierpinski ring extensions (Theorem 3). In the second part we give a characterization of an $a$-non-connected pre-image as an absolutely countably non-connected covering (Theorem 4). Descriptions are also given of Baire and Borel pre-images generated by the classical Baire and Borel measurable extensions (Theorem 5).