On $\beta^{*}-\mathrm{I}$-open Sets and $e^{*}-\mathrm{I}$-open Sets in Ideal Topological Spaces

In this study, we explore and delineate several characteristics of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\beta^{*}-\mathrm{I}$</tex> -open sets, and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$e^{*}-\mathrm{I}$</tex> -open sets within ideal topological spaces. Additionally, we establish connections between <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$e^{*}$</tex> -I-open sets, and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\beta^{*}$</tex> -I-open sets in these spaces. Ultimately, we achieve a decomposition of continuity, enhancing the understanding of how these sets interact and contribute to the broader topological framework.