  Directed spaces are natural topological extensions of dcpos in domain theory
and form a cartesian closed category. We will show that the D-completion of
free algebras over a Scott space $\Sigma L$, on the context of directed spaces,
are exactly the free dcpo-algebras over dcpo $L$, which reveals the close
connection between directed powerspaces and powerdomains. By this result, we
provide a topological representation of upper, lower and convex powerdomains of
dcpos uniformly.
