4 research outputs found
-quasicontinuous spaces
In this paper, as a common generalization of -continuous spaces and
-quasicontinuous posets, we introduce the concepts of
-quasicontinuous spaces and -convergence of nets for
arbitrary topological spaces by the cuts. Some characterizations of
-quasicontinuity of spaces are given. The main results are: (1) a space
is -quasicontinuous if and only if its weakly irreducible topology is
hypercontinuous under inclusion order; (2) A space is
-quasicontinuous if and only if the -convergence in
is topological
Countably QC
As a generalization of countably C-approximating posets, the concept of countably QC-approximating posets is introduced. With the countably QC-approximating property, some characterizations of generalized completely distributive lattices and generalized countably approximating posets are given. The main results are as follows: (1) a complete lattice is generalized completely distributive if and only if it is countably QC-approximating and weakly generalized countably approximating; (2) a poset L having countably directed joins is generalized countably approximating if and only if the lattice σcLop of all σ-Scott-closed subsets of L is weakly generalized countably approximating
Quasiexact posets and the moderate meet-continuity
The study of weak domains and quasicontinuous domains leads to the
consideration of two types generalizations of domains. In the current paper, we
define the weak way-below relation between two nonempty subsets of a poset and
quasiexact posets. We prove some connections among quasiexact posets,
quasicontinuous domains and weak domains. Furthermore, we introduce the weak
way-below finitely determined topology and study its links to Scott topology
and the weak way-below topology first considered by Mushburn. It is also proved
that a dcpo is a domain if it is quasiexact and moderately meet continuous with
the weak way-below relation weakly increasing
D-completions and the d-topology
In this article we give a general categorical construction via reflection functors for various completions of T0-spaces subordinate to sobrification, with a particular emphasis on what we call the D-completion, a type of directed completion introduced by Wyler [O. Wyler, Dedekind complete posets and Scott topologies, in: B. Banaschewski, R.-E. Hoffmann (Eds.), Continuous Lattices Proceedings, Bremen 1979, in: Lecture Notes in Mathematics, vol. 871, Springer Verlag, 1981, pp. 384-389]. A key result is that all completions of a certain type are universal, hence unique (up to homeomorphism). We give a direct definition of the D-completion and develop its theory by introducing a variant of the Scott topology, which we call the d-topology. For partially ordered sets the D-completion turns out to be a natural dcpo-completion that generalizes the rounded ideal completion. In the latter part of the paper we consider settings in which the D-completion agrees with the sobrification respectively the closed ideal completion. © 2008 Elsevier B.V. All rights reserved