On the lattice of e-open sets and od-compact spaces

[EN] In this paper, we investigate od-compact spaces, a natural generalization of e-compact spaces (hence compact spaces), and study their interplay with the lattice-theoretic structure of e-open and open dense subsets, denoted by E(X) and OD(X), respectively. We first provide an algebraic characterization of od-compactness and e-compactness in terms of essential and Baer ideals of C(X), respectively. We then observe the relationships between od-compactness, connectedness, hyperconnectedness, and compactness, and examine the behavior of these notions under continuous open maps and products. Furthermore, we analyze the lattice properties of E(X) and OD(X), showing that they are closely related to the lattice of Baer and essential ideals of C(X), respectively. Several structural properties of these lattices, such as compactness, connectedness, and co-normality, are discussed, along with their preservation under homeomorphisms and ring isomorphisms. Finally, we establish isomorphisms between these lattices and their counterparts in the Stone-?ech compactification.