The analytical premanifolds and the complex premanifolds, defined and studied by the author in two previous papers, are called in this paper general differential spaces. The general differential spaces are also called \(K\)-differential spaces (\(K\)-d.s.), where \(K=\mathbb{R}\) or \(K=\mathbb{C}\). For any indexed set of mappings, a \(K\)-d.s. induced by this indexed set is defined and characterized by a universal property. Similarly, a \(K\)- d.s. coinduced by any indexed set of mappings is defined and universally characterized. It is proved that the topology of the induced (resp. coinduced) \(K\)-d.s. is the induced (resp. coinduced) topology by the indexed set of the mappings.