We study algebraic varieties
parametrized by topological spaces and enlarge the domain of Lawson homology
and morphic cohomology to this category. We prove a Lawson suspension
theorem and a splitting theorem. A version of the Friedlander–Lawson moving
lemma is obtained to prove a duality theorem between Lawson homology
and morphic cohomology for smooth semi-topological projective varieties.
K-groups for semi-topological projective varieties and Chern classes are also
constructed.