Through convergences of directed subsets of a T0 space with specialization order, we introduce a new topological space——directed space. We show that these directed spaces have some similar properies satisfied by the directed complete posets (dcpos). Particularly, we show that a category of directed space and continuous functions has following properties: (i) it is a co reflective full subcategory of the category of T0 space; (ii) it is cartesian closed and contains the category of dcpos with Scott topologies as a proper subcategory.
