In this paper, we study the approximation relations derived by directed sets on \(T_{0}\) -spaces. It is shown that the approximation relation can be regarded as a suitable generalization of the classical way below relation \(\ll \) and Erné’s relation \(\ll _{2}\) on posets to the more general setting of \(T_0\) -spaces. Using the approximation relation, we introduce and investigate approximation spaces on \(T_0\) -spaces and give some characterizations of approximation spaces by the MD-open sets. It is proved that a \(T_0\) -space X is a d-space iff its MD-space is a d-space, and X is an approximation space iff its MD-space is a c-space. Therefore, X is a c-space iff X is both an approximation space and an MD-space iff X is a locally hypercompact approximation space. So approximation spaces can be seen as a generalization of continuous spaces and both concepts are united in an MD-space.
