Continuity and Directed Completion of Topological Spaces

In this paper we introduce the notions of continuity, meet continuity and quasicontinuity of Delta-spaces, where a Delta-space is a monotone determined, weak monotone convergence space. We prove that: a Delta-space (X, O(X)) is continuous iff the topology lattice O(X) is completely distributive; a Delta-space (X, O(X)) is meet continuous iff O(X) is a complete Heyting algebra; a Delta-space (X, O(X)) is continuous iff it is both meet continuous and quasicontinuous. The concepts of continuity, s(2)-continuity, O-continuity, strong continuity of posets are shown to be special cases of the continuity of Delta-spaces. We also introduce a type of directed completion of Delta-spaces, called Scott completion. For each Delta-space (X, O(X)), the topology lattice O(X) is isomorphic to the topology lattice of its Scott completion. The D-completion, D-theta-completion and D-s2 -completion of posets are included in the Scott completion.