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.