The ${\omega }^{\mathrm{♯}}$-operator in ideal topological spaces and its associated topology

In this paper, we introduced a new set-theoretic operator $(\cdot {)}_{\omega }^{\mathrm{♯}}$ in the framework of ideal topological spaces and investigated its fundamental properties, including its connections with the classical $\mathrm{♯}$-operator and the $\omega$-local function. Using this operator, we defined a closure-type operator ${\mathrm{C}\mathrm{l}}_{\omega }^{\mathrm{♯}}$ and showed that it satisfies the Kuratowski closure axioms. Consequently, a topology ${\mathcal{T}}_{\omega }^{\mathrm{♯}}$ was obtained, which is strictly finer than the topology induced by the $\mathrm{♯}$-operator. Furthermore, the structural relationships among these topologies were examined, and some applications of the ${\omega }^{\mathrm{♯}}$-operator were presented. Finally, we introduced the notions of ${\omega }^{\ast }$-continuity and ${\omega }^{\mathrm{♯}}$-continuity, investigated their relationship, and established a new decomposition of continuity. We also compared these notions with related concepts such as $\omega$-continuity and $\mathrm{♯}$-continuity.