Some covering properties and connectedness via Gα-open sets

The novel ideas in grill topological spaces are explored by applying them to the setting of specified sets, where the -open sets are defined. The set's attributes are investigated, and examples and properties related to this set are provided. The set includes a variety of functions, such as open, closed, and continuous functions. This is the first step in a series of analyses of various topological features of this collection. By studying the new concept G(alpha) -open set and it is properties we created a new concept can be named as connectedness via G(alpha)-open sets and some covering properties.