An Introduction to Mixed $\mathcal{H}\big(\theta(\mu, \nu)\big)$-Open Sets Generated by Hereditary Classes in Generalized Topological Spaces

In [1], Kim and Min introduced the operation $\gamma_{\ast}$ and $\mathcal{H}(\theta)$-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, $\mu$ and $\nu$, along with a hereditary class $\mathcal{H}$. Specifically, we introduce and investigate the mixed operation $\gamma_{*}(\mu, \nu)$ (denoted briefly as $\gamma_{*}(\mu, \nu)$) and the mixed $\mathcal{H}(\theta(\mu, \nu))$-open sets (denoted as $\mathcal{H}(\theta(\mu, \nu))$-open sets). We explore the interrelationships between $\gamma_{*}(\mu, \nu)$, $\gamma_{\ast}$, and the $\mu$-closure, as well as the connections between $\mathcal{H}(\theta(\mu, \nu))$-open sets, $\theta(\mu, \nu)$-open sets, and $\mu$-open sets. Additionally, we define the concepts of $\mathcal{H}r(\mu, \nu)$-regular open sets and $\mathcal{H}(\mu, \nu)$-regular open sets. Finally, we examine properties and characterizations of $\mathcal{H}(\theta(\mu, \nu))$-open sets in terms of $\mathcal{H}r(\mu, \nu)$-regular open sets and $\mathcal{H}(\mu, \nu)$-regular open sets.