In [R. Heckmann, K. Keimel, Quasicontinuous Domains and the Smyth Powerdomain, Electronic Notes in Theoretical Computer Science 298 (2013), 215–232], Heckmann and Keimel proved that a dcpo P is quasicontinuous iff the poset FinP of nonempty finitely generated upper sets ordered by reverse inclusion is continuous. We generalize this result to general topological spaces in this paper. More precisely, for any space and , we construct a topology generated by the basic open subsets . It is shown that a space is a hypercontinuous lattice iff is a completely distributive lattice. In particular, we prove that if a poset P satisfies property DINTop, then P is quasi-hypercontinuous iff FinP is hypercontinuous.