  We investigate two approximation relations on a T0 topological space, the
n-approximation, and the d-approximation, which are generalizations of the
way-below relation on a dcpo. Different kinds of continuous spaces are defined
by the two approximations and are all shown to be directed spaces. We show that
the continuity of a directed space is very similar to the continuity of a dcpo
in many aspects, which indicates that the notion of directed spaces is a
suitable topological extension of dcpos.The main results are: (1) A topological
space is continuous iff it is a retract of an algebraic space;(2) a directed
space X is core compact iff for any directed space Y, the topological product
is equal to the categorical product in DTop of X and Y respectively;(3) a
directed space is continuous (resp., algebraic, quasicontinuous,
quasialgebraic) iff the lattice of its closed subsets is continuous (resp.,
algebraic, quasicontinuous, quasialgebraic).
