Abstract
AbstractThe notions ofos{o}_{s}-convergence andS∗{S}^{\ast }-doubly quasicontinuous posets are introduced, which can be viewed as common generalizations of Birkhoff’s order-convergence andS∗{S}^{\ast }-doubly continuous posets, respectively. We first consider the relationship betweenos{o}_{s}-convergence and B-topology and show that the topology induced byos{o}_{s}-convergence according to the standard topological approach is the B-topology precisely. Then, the topological characterization for theS∗{S}^{\ast }-doubly quasicontinuity is presented. It is proved that a poset isS∗{S}^{\ast }-doubly quasicontinuous iff the poset equipped with the B-topology is locally hyperclosed iff the lattice of all B-open subsets of the poset is hypercontinuous. Finally, the order theoretical condition for theos{o}_{s}-convergence being topological is given and the complete regularity of B-topology onS∗{S}^{\ast }-doubly quasicontinuous posets is explored.
Highlights
Based on the o-convergence, the order topology on posets has been defined by Birkhoff [1] according to the standard topological approach
This topology played an important role in searching the order-theoretical condition for the o-convergence being topological. This fact can be demonstrated in the work of Zhao and Wang in [9]. They considered the relationship between the B-topology and Bi-Scott topology and showed that a poset P, which satisfies condition (∗), is doubly continuous if and only if o-convergence in P is topological with respect to the B-topology on P
The main goal of this paper is to explore the relationship between B-topology and S∗-double quasicontinuity, a generalized form of S∗-double continuity
Summary
The concept of order-convergence (o-convergence, for short) in partially ordered sets was introduced by Birkhoff [1], Frink [2] and Mcshane [3] and studied by Mathews and Anderson [4], Wolk [5] and Olejček [6]. [7] A poset P is said to be S-doubly continuous if for every x ∈ P, the sets ⇓Sx and ⇈Sx are directed and filtered, respectively, and sup⇓Sx = x = inf ⇈Sx. Definition 1.5. A net (xi)i∈I in P is said to os-converge to x ∈ P if there exists a directed family x ⊆ 0(↓x) and a filtered family x ⊆ 0(↑x) such that (Q1) ⋂{↑M : M ∈ x} = ↑x and ⋂{↓N : N ∈ x} = ↓x. We provide an approach to construct a special kind of os-convergent net in a given poset. The proof can be completed by checking that the directed family x and the filtered family x satisfy Conditions (Q1) and (Q2) of Definition 2.1
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