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