Upper powerdomains of quasicontinuous dcpos

The upper powerdomain of an arbitrary dcpo L can be constructed by the Scott completion of the directed upper powerspace of L . By means of this result, we investigate the upper powerdomains of quasicontinuous domains. The main results are: (1) The upper powerdomain P u ( L ) of a quasicontinuous domain L is quasicontinuous iff the directed upper powerspace of L is endowed with the Scott topology, iff P u ( L ) is isomorphic to Q ( L ) , the semilattice of nonempty Scott compact saturated subsets of L . (2) There exists a quasicontinuous dcpo whose upper powerdomain is not quasicontinuous. (3) The upper powerdomain of a strongly quasicontinuous domain L is isomorphic to Q ( L ) .