In this paper, we introduced a new set-theoretic operator $ (\cdot)^{\sharp}_{\omega} $ in the framework of ideal topological spaces and investigated its fundamental properties, including its connections with the classical $ \sharp $-operator and the $ \omega $-local function. Using this operator, we defined a closure-type operator $ \mathrm{Cl}^{\sharp}_{\omega} $ and showed that it satisfies the Kuratowski closure axioms. Consequently, a topology $ \mathcal{T}^{\sharp}_{\omega} $ was obtained, which is strictly finer than the topology induced by the $ \sharp $-operator. Furthermore, the structural relationships among these topologies were examined, and some applications of the $ \omega^\sharp $-operator were presented. Finally, we introduced the notions of $ \omega^\ast $-continuity and $ \omega^\sharp $-continuity, investigated their relationship, and established a new decomposition of continuity. We also compared these notions with related concepts such as $ \omega $-continuity and $ \sharp $-continuity.
Citation: Abdo Qahis, Mohd Salmi Md Noorani. The $ \omega^{\sharp} $-operator in ideal topological spaces and its associated topology[J]. AIMS Mathematics, 2026, 11(6): 17039-17061. doi: 10.3934/math.2026698
In this paper, we introduced a new set-theoretic operator $ (\cdot)^{\sharp}_{\omega} $ in the framework of ideal topological spaces and investigated its fundamental properties, including its connections with the classical $ \sharp $-operator and the $ \omega $-local function. Using this operator, we defined a closure-type operator $ \mathrm{Cl}^{\sharp}_{\omega} $ and showed that it satisfies the Kuratowski closure axioms. Consequently, a topology $ \mathcal{T}^{\sharp}_{\omega} $ was obtained, which is strictly finer than the topology induced by the $ \sharp $-operator. Furthermore, the structural relationships among these topologies were examined, and some applications of the $ \omega^\sharp $-operator were presented. Finally, we introduced the notions of $ \omega^\ast $-continuity and $ \omega^\sharp $-continuity, investigated their relationship, and established a new decomposition of continuity. We also compared these notions with related concepts such as $ \omega $-continuity and $ \sharp $-continuity.
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Abdo Qahis, Mohd Salmi Md Noorani. The $ \omega^{\sharp} $-operator in ideal topological spaces and its associated topology[J]. AIMS Mathematics, 2026, 11(6): 17039-17061. doi: 10.3934/math.2026698
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