In this paper, we investigate the property of core compactness of ordered topological spaces. Particularly, we give a series of characterizations of the core compactness for directed spaces. Several results obtained in this paper are closely related to a long-standing open problem in Open problems in Topology (J. van Mill, G. M. Reed Eds., North-Holland, 1990): Which distributive continuous lattice's spectrum is exactly a sober locally compact Scott space?

Directed space was defined by Hui Kou in 2014 ^[21], which is equivalent to $T_0$ monotone determined space. Its main purpose is to build an extended framework for domain theory. In this paper, we study the category of c-spaces which is a subcategory of directed spaces. The main results are: (1) we will describe c-spaces using a new definition, which give us the convenience to construct new classes of spaces; (2) we give some conditions such that categorical products and topological products agree in $\mathbf{D}\mathbf{t}\mathbf{o}\mathbf{p}$; (3) the category of c-spaces is not Cartesian closed; (4) we define a new class of spaces, namely, FS-spaces, which forms a Cartesian closed category.