Generalized Topological Structures Induced by $H^{*}$-Open Sets and Their Separation Properties
In this paper, we investigate the family of $H^{*}$-open sets from the perspective of generalized topology and examine the structural and separation properties arising from it. For a topological space $(\mathbb{X},τ)$, the family of all $H^{*}$-open subsets of $\mathbb{X}$ is considered as a natural candidate for an $H^{*}$-generated generalized topology. We establish conditions under which this family forms a strong generalized topology and identify additional assumptions that allow the resulting structure to be treated as an ordinary topology. This distinction provides a systematic framework for studying $H^{*}$-open sets without imposing finite-intersection properties in advance. Within this framework, we investigate the corresponding $H^{*}$-interior and $H^{*}$-closure operators and analyze separation properties determined by the $H^{*}$-open structure. When the $H^{*}$-generated family forms a topology, the resulting associated space is used to relate generalized $H^{*}$-separation properties to their classical topological counterparts. We further examine compactness-type properties, subspace behaviour, and preservation under suitable mappings. Examples are included to illustrate how the structure generated by $H^{*}$-open sets may exhibit separation behaviour different from that of the original topology. The results provide a unified setting for connecting $H^{*}$-open-set theory with generalized topological structures and offer a basis for further investigation of separation and covering properties in this framework.