The ω♯-Operator in Ideal Topological Spaces and Its Associated Topology

In this paper, we introduce a new set-theoretic operator $(cdot)^{sharp}_{omega}$ in the framework of ideal topological spaces and investigate its fundamental properties, including its connections with the classical $sharp$-operator and the $omega$-local function. Using this operator, we define a closure-type operator $mathrm{Cl}^{sharp}_{omega}$ and show that it satisfies the Kuratowski closure axioms. Consequently, a topology $mathcal{T}^{sharp}_{omega}$ is obtained, which is strictly finer than the topology induced by the $sharp$-operator. Furthermore, the structural relationships among these topologies are examined, and some applications of the $omega^sharp$-operator are presented. Finally, we introduce the notions of $omega^ast$-continuity and $omega^sharp$-continuity, investigate their relationship, and establish a new decomposition of continuity. We also compare these notions with related concepts such as $ast$-continuity and $sharp$-continuity.