We study here $T_{0}$-$Q$-bitopological spaces and sober $Q$-bitopological spaces and their relationship with two particular Sierpinski objects in the category of $Q$-bitopological spaces. The epireflective hulls of both these Sierpinski objects in the category of $Q$-bitopological spaces turn out to be the category of $T_0$-$Q$-bitopological spaces. We show that only one of these Sierpinski ob...

A poset model of a topological space X is a poset P together with a homeomorphism φ : X−→Max(P ) (Max(P ) is the subspace of the Scott space ΣP consisting of maximal points of P ). In [11] (also in [2]), it was proved that every T1 space has a bounded complete algebraic poset model. It is, however still unclear whether each T1 space has a dcpo model. In this paper we give a positive answer to t...

Abstract The study of the sobriety Scott spaces has got a relatively long history in domain theory. Lawson and Hoffmann independently proved that space every continuous directed complete poset (usually called domain) is sober. Johnstone constructed first whose non-sober. Soon after, Isbell gave lattice with non-sober space. Based on Isbell’s example, Xu, Xi, Zhao showed there even Heyting algeb...

In [4], Erné relaxed the concept of sobriety in order to extend theory sober spaces and locally hypercompact situations where directed joins were missing, introduced discussed three kinds non-sober spaces: cut spaces, weakly quasisober spaces. [24], other namely DC space, RD space WD investigated. All these six classes are strictly larger than that this paper, it is shown category all not refle...

• A uniform approach to d -spaces, sober spaces and well-filtered is provided. The theory of (super) H-sober established. If H has property M, then the function space super equipped with topology pointwise convergence H-sober. a T 0 X iff its Smyth power P S ( ) category all reflective in Top . if M. In this paper, we provide spaces, develop general framework for dealing these spaces. concepts ...

The theme of this paper is the extension of continuous valuations on the lattice of open sets of a T0-space to Borel measures. A general extension principle is derived that provides a unified approach to a variety of extension theorems including valuations that are directed suprema of simple valuations, continuous valuations on locally compact sober spaces, and regular valuations on coherent so...