Connections between the Lebesgue extension and the Borel extension of the first class, and between the preimages corresponding to them

A new algebraic structure of a $c$-ring with refinement and a new topological structure of an $a$-space with cover are introduced. On the basis of them the notions of divisible hulls and surrounded coverings of certain types are introduced. With the help of these notions the Lebesgue extension $C\rightarrowtail L_\mu$ and the Borel extension $C\rightarrowtail BM_1$ of the first class are given a ring characterization as divisible hulls of a certain type (Theorem 1); preimages of maximal ideals of these extensions are given a topological characterization as surrounded coverings of a certain type (Theorem 2).