Continuity and Directed Completion of Topological Spaces

Abstract

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