This work introduces a new class of sets, called h-open sets, defined through an operator γ and denoted by hγ-open sets, within the framework of a topological structure. The analysis focuses on the connection between these sets and those defined as γ-open, identifying precise criteria under which both families coincide. We also investigate properties of hγ-open sets, such as stability under unrestricted set unions and intersections involving only a limited number of components. Furthermore, we explore the connections between various notions of γ-open sets and h-open sets. Finally, we revisit a lemma from the literature that attempts to characterize h-open sets and provide counterexamples, as well as identify specific conditions under which the lemma holds true.