  Directed spaces are natural topological extensions of dcpos in domain theory
and form a cartesian closed category. In order to model nondeterministic
semantics, the power structures over directed spaces were defined through the
form of free algebras. We show that free algebras over any directed space exist
by the Adjoint Functor Theorem. An c-space (resp.\ b-space) can be
characterized as a continuous (resp.\ algebraic) directed space. We show that
continuous spaces are just all retracts of algebraic spaces by means of
topological ideals, which are generalizations of the rounded ideals. Moreover,
by categorical methods, we show that the carrier spaces of free algebras over
continuous (resp.\ algebraic) spaces are still continuous (resp.\ algebraic)
spaces.
