On -open sets

This study investigates new concepts in grill topological spaces by employing specified sets in which the α-open sets are defined. Many scholars, such as Choquet, who is considered the first to lay the foundation stone for the concept of a grill and formulate its definition, were interested in studying it. After that, many attempts have been made to study the properties associated with this concept and to understand the relationships among these properties. There are several types of grill topological spaces, including discrete and cofinite topologies. Properties of this set and certain relationships are studied, as well as examining a group of functions, like open, closed, and continuous functions, determining their relation with each other and giving examples and properties associated with this set. This shall serve as the beginning of examining numerous topological properties with this set.