Malaysian Journal of Mathematical Sciences, December 2025, Vol. 19, No. 4
Novel Classes of Sets in Ideal Čech Closure Spaces
Hosny, R. A., Amin, W. M., and Abu-Donia, H. M.
Corresponding Email: waheedamin90@gmail.com
Received date: 12 January 2025
Accepted date: 28 May 2025
Abstract:
Relaxing the idempotent condition in closure operations leads to a generalized framework for topological spaces, offering significant advantages in modeling proximity. This work explores the extension of classical closure concepts within the framework of ideal Čech closure spaces. This extension establishes the idea of generalized Čech closed sets relative to an ideal $\mathbb{L}$, known as $\mathbb{L}$g-Čclosed sets. This definition aims to expand the theoretical and practical utility of closure notions by incorporating ideals, allowing for more comprehensive, and applicable frameworks in mathematical scopes. Additionally, the research investigates the properties of these $\mathbb{L}$g-Čclosed sets, highlighting their role in offering new insights into topological structures and closure operations in settings where traditional closure conditions are insufficient. Also, we have developed an algorithm for identifying $\mathbb{L}$g-Čclosed sets, which provides a figurative representation to enhance understanding of the underlying concepts and computational steps. These novel ideas lead to the introduction of new kinds, namely $\Omega^{\mathbb{L}}$-sets and $\mho^{\mathbb{L}}$-sets, within Čech closure spaces. Specifically, the structure $(\Upsilon, \Omega^{\mathbb{L}})$ forms a convex closure space, and several fundamental properties related to these sets are established. Furthermore, by providing counterexamples to one-sided theorems, the research demonstrates situations where the converse does not hold, thus spotlighting the complexities and limitations of these generalized frameworks.
Keywords: Čech closure space; ideals; $\mathbb{L}$g-Čclosed and $\mathbb{L}$g-Čopen sets; $\Omega^{\mathbb{L}}$-sets and $\mho^{\mathbb{L}}$-sets
