The paper by Olav Njåstad explores the structure and properties of $\alpha$-sets and $\beta$-sets in topological spaces. $\alpha$-sets are defined as sets $A$ for which $A^{0-0} \supset A$, while $\beta$-sets are defined as sets $B$ for which $B^{0-} \supset B$. The paper investigates the relationships between these sets and their topologies, showing that topologies determining the same class of $\alpha$-sets also determine the same class of $\beta$-sets, and vice versa. It is shown that the class of $\beta$-sets forms a topology if and only if the original topology is extremally disconnected, and the class of $\alpha$-sets always forms a topology. The paper also discusses the convexity of $\alpha$-classes in the ordering by inclusion and provides conditions for a topology to be the coarsest in its $\alpha$-class. Additionally, it examines the continuous mappings into regular spaces and the properties of quasi-regular topologies. The paper concludes with examples and corollaries to illustrate the theoretical results.The paper by Olav Njåstad explores the structure and properties of $\alpha$-sets and $\beta$-sets in topological spaces. $\alpha$-sets are defined as sets $A$ for which $A^{0-0} \supset A$, while $\beta$-sets are defined as sets $B$ for which $B^{0-} \supset B$. The paper investigates the relationships between these sets and their topologies, showing that topologies determining the same class of $\alpha$-sets also determine the same class of $\beta$-sets, and vice versa. It is shown that the class of $\beta$-sets forms a topology if and only if the original topology is extremally disconnected, and the class of $\alpha$-sets always forms a topology. The paper also discusses the convexity of $\alpha$-classes in the ordering by inclusion and provides conditions for a topology to be the coarsest in its $\alpha$-class. Additionally, it examines the continuous mappings into regular spaces and the properties of quasi-regular topologies. The paper concludes with examples and corollaries to illustrate the theoretical results.