New Type of Grill Topological Spaces

The primary objective of this work pertains to the examination of a new class of grill topological spaces denominated as AE-space. This nomenclature is rooted in the delineation of said spaces upon a non-empty set, upon which a myriad of topological structures is imposed via an arbitrary grill. The construction of these grill topologies precedes the establishment of our nascent topological construct through the intersection of these individualized topologies. In this discourse, we introduce and rigorously define the notions of AE-open sets and AE-closed sets. Furthermore, we establish the result that the AE-space exhibits the fundamental attributes of a topological space, assuming the absence of pairwise disjoint non-empty sets within the intersection of the grill topologies. Illustrative instances of such AE-spaces are furnished, accompanied by a comprehensive demonstration of their pertinent properties.