The paper by Olav Njåstad, "On Some Classes of Nearly Open Sets," published in the Pacific Journal of Mathematics in 1965, investigates the structure and properties of α-sets and β-sets in topological spaces. These sets are defined based on their relationship with open sets and their closures and interiors. An α-set is one for which $ A^{0-0} \supset A $, while a β-set satisfies $ B^{0} \supset B $. The paper shows that the classes of α-sets and β-sets are closely related, and that topologies determining the same α-structure also determine the same β-structure, and vice versa. It is shown that the class of β-sets forms a topology if and only if the original topology is extremely disconnected. The class of α-sets always forms a topology, and α-topologies are exactly those where all nowhere dense sets are closed. The class of all topologies that determine the same α-sets is convex in the ordering by inclusion, with the α-topology being the finest member. Most common topologies are the coarsest members of their respective classes. The paper also explores the relationship between α-sets and β-sets, showing that every β-set is the union of an open set and a nowhere dense set, and that every α-set can be expressed as the difference between an open set and a nowhere dense set. These properties lead to corollaries about the structure of α-topologies and their relationship to Baire-topologies. The paper further discusses the characterization of α-structures in terms of β-structures and vice versa, and provides a characterization of α-topologies as topologies where the difference between an open set and a nowhere dense set is again an open set. It also shows that α-equivalent topologies determine the same class of continuous mappings into arbitrary regular spaces and the same class of quasicontinuous mappings into arbitrary topological spaces. The paper concludes with a discussion of the order structure of α-classes, showing that every α-class is convex in the set of topologies on E ordered by inclusion. It also provides conditions for an α-topology to be the only topology of its class, and discusses the properties of quasi-regular topologies and their relationship to α-classes. The paper also includes an example showing that an α-topology may not be quasi-regular at any point, and that an α-class may not always possess a coarsest topology.The paper by Olav Njåstad, "On Some Classes of Nearly Open Sets," published in the Pacific Journal of Mathematics in 1965, investigates the structure and properties of α-sets and β-sets in topological spaces. These sets are defined based on their relationship with open sets and their closures and interiors. An α-set is one for which $ A^{0-0} \supset A $, while a β-set satisfies $ B^{0} \supset B $. The paper shows that the classes of α-sets and β-sets are closely related, and that topologies determining the same α-structure also determine the same β-structure, and vice versa. It is shown that the class of β-sets forms a topology if and only if the original topology is extremely disconnected. The class of α-sets always forms a topology, and α-topologies are exactly those where all nowhere dense sets are closed. The class of all topologies that determine the same α-sets is convex in the ordering by inclusion, with the α-topology being the finest member. Most common topologies are the coarsest members of their respective classes. The paper also explores the relationship between α-sets and β-sets, showing that every β-set is the union of an open set and a nowhere dense set, and that every α-set can be expressed as the difference between an open set and a nowhere dense set. These properties lead to corollaries about the structure of α-topologies and their relationship to Baire-topologies. The paper further discusses the characterization of α-structures in terms of β-structures and vice versa, and provides a characterization of α-topologies as topologies where the difference between an open set and a nowhere dense set is again an open set. It also shows that α-equivalent topologies determine the same class of continuous mappings into arbitrary regular spaces and the same class of quasicontinuous mappings into arbitrary topological spaces. The paper concludes with a discussion of the order structure of α-classes, showing that every α-class is convex in the set of topologies on E ordered by inclusion. It also provides conditions for an α-topology to be the only topology of its class, and discusses the properties of quasi-regular topologies and their relationship to α-classes. The paper also includes an example showing that an α-topology may not be quasi-regular at any point, and that an α-class may not always possess a coarsest topology.