Abstract: We used an ideal convergence structure to solve the characterization problem of directed topology, and provided necessary and  sufficient conditions for the topological transformation of ideal S limits and ideal generalized S limits. The results show that the directed topology, the ideal S limit topology and the ideal generalized S limit topology are the same  in T₀ topological spaces.  The ideal S convergence in a directed space is topological if and only if it is a c-space. The ideal generalized S convergence in a directed space is topological if and only if it is a locally strongly compact space.

Key words: ideal S limit, ideal generalized S limit, c-space, locally strongly compact space, directed topology