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THE SCOTT TOPOLOGY AND LAWSON TOPOLOGY ON A Z-QUASICONTINUOUS DOMAIN

For a general subset system Z, the concept of a Z-quasicontinuous domain is introduced, and the following results are proved: (1) a Z-complete poset P is Z-quasicontinuous if and only if the complete lattice of all Z-Scott open subsets of P is hypercontinuous, (2) the Z-Scott topology σZ(P) on a Z-quasicontinuous poset P is Sober if and only if σZ(P) has the Rudin property, and (3) a Z-quasicontinuous poset P endowed with the Z-Lawson topology λZ(P) is a pospace. Furthermore, if all upper Z-Lawson open subsets of P are Z-Scott open and all lower Z-Lawson open subsets of P are open in (P, w(P)), then (P, λZ(P)) is a strictly completely regular ordered space.