The idea of compactness considered as an essential conception not only in general topology but also in other superior branches of mathematics. Therefore, in this article new notion of compactness namely; h-compactness in the setting of topological spaces has been presented based on h-open sets. Several basic properties and important examples related to this kind of spaces have been discussed. On the other hand, new type of extended mapping namely; h-compact mapping has been defined via the notion of h-open sets. Moreover, in the process we introduce another type of extended mappings namely; strongly h-compact mapping, as well the relationships between these kinds of mappings are investigated