Abstract:

The purpose of this paper is to extend the study of generalized closed sets in leras topological spaces to the ideal setting. Using the leras local function and leras \( \star \)-closure operator associated with an ideal, we introduce and investigate two new classes of sets, called \( ideal\ leras\ generalized\ closed\ sets \) (\( I_{lrg} \)-closed sets) and \( ideal\ leras\ generalized\ open\ sets \) (\( I_{lrg} \)-open sets), in ideal leras topological spaces. We show that every \( lrg \)-closed set is \( I_{lrg} \)-closed, and that every \( I^\star \)-closed set is \( I_{lrg} \)-closed, while the converses fail in general, thereby placing the new classes correctly within the existing hierarchy of sets in leras and ideal leras topological spaces. Several characterizations of \( I_{lrg} \)-closed sets are obtained, including formulations in terms of the leras \( \star \)-closure operator, the leras closures of singletons, and the non-existence of nonempty \( Ir \)-closed sets in certain associated difference sets. Additional properties, such as finite unions of \( I_{lrg} \)-closed sets being \( I_{lrg} \)-closed and sufficient conditions under which \( I_{lrg} \)-closed sets coincide with \( lrg \)-closed sets, are also established. Appropriate examples are constructed throughout to illustrate the developed concepts and to demonstrate that the implications obtained cannot, in general, be reversed.