Towards a Unified Framework for Bioperations on $e^{\star }$-Open Sets in Topological Spaces

In this paper, we introduce the concept of $e_{[\gamma ,{\gamma }^{\mathrm{\prime }}]}^{\star }$-open sets in topological spaces and examine their properties in detail. Additionally, we propose a new class of functions, termed $(e_{[\gamma ,{\gamma }^{\mathrm{\prime }}]}^{\star },e_{[\beta ,{\beta }^{\mathrm{\prime }}]}^{\star })$-continuous functions, and explore their fundamental characteristics.