An Introduction to Mixed $\mathcal{H}\big(θ(μ, u)\big)$-Open Sets Generated by Hereditary Classes in Generalized Topological Spaces (European Journal of Pure and Applied Mathematics, 2025)
In [1], Kim and Min introduced the operation $γ_{*}$ and $\mathcal{H}(θ)$-open sets within the context of generalized topological spaces, utilizing a hereditary class $\mathcal{H}$. In this study, we extend these concepts by employing two generalized topologies, $μ$ and $ u$, along with a hereditary class $\mathcal{H}$. Specifically, we introduce and investigate the mixed operation $γ_{*}(μ, u)$ (denoted briefly as $γ_{*}(μ, u)$) and the mixed $\mathcal{H}(θ(μ, u))$-open sets (denoted as $\mathcal{H}(θ(μ, u))$-open sets). We explore the interrelationships between $γ_{*}(μ, u)$, $γ_{*}$, and the $μ$-closure, as well as the connections between $\mathcal{H}(θ(μ, u))$-open sets, $θ(μ, u)$-open sets, and $μ$-open sets. Additionally, we define the concepts of $\mathcal{H}r(μ, u)$-regular open sets and $\mathcal{H}(μ, u)$-regular open sets. Finally, we examine properties and characterizations of $\mathcal{H}(θ(μ, u))$-open sets in terms of $\mathcal{H}r(μ, u)$-regular open sets and $\mathcal{H}(μ, u)$-regular open sets.
Keywords: $F$-open; $FT_i$ ($i = 0, 1, 2, 3, 4$); $F$-regular; $F$-normal