Lattice-valued Scott topology on dcpos

2015 ◽  
Vol 27 (4) ◽  
pp. 516-529
Author(s):  
WEI YAO
Keyword(s):  
Topological Space ◽  
Continuous Semigroup ◽  
Scott Topology ◽  
Completely Distributive ◽  
Sober Space ◽  
Truth Value

This paper studies the fuzzy Scott topology on dcpos with a *-continuous semigroup (L, *) as the truth value table. It is shown that the fuzzy Scott topological space on a continuous dcpo is an ιL-sober space. The fuzzy Scott topology is completely distributive iff L is completely distributive and the underlying dcpo is continuous. For (L, *) being an integral quantale, semantics of L-possibility of computations is studied by means of a duality.

scholarly journals Scott convergence and fuzzy Scott topology on L-posets

2017 ◽  
Vol 15 (1) ◽  
pp. 815-827 ◽  
Author(s):  
Hongping Liu ◽  
Ling Chen
Keyword(s):  
Topological Space ◽  
Scott Topology ◽  
Convergence Structure

Abstract We firstly generalize the fuzzy way-below relation on an L-poset, and consider its continuity by means of this relation. After that, we introduce a kind of stratified L-generalized convergence structure on an L-poset. In terms of that, L-fuzzy Scott topology and fuzzy Scott topology are considered, and the properties of fuzzy Scott topology are discussed in detail. At last, we investigate the Scott convergence of stratified L-filters on an L-poset, and show that an L-poset is continuous if and only if the Scott convergence on it coincides with the convergence with respect to the corresponding topological space.

scholarly journals A new convergence inducing the SI-topology

Filomat ◽  
2018 ◽  
Vol 32 (17) ◽  
pp. 6017-6029 ◽  
Author(s):  
Hadrian Andradi ◽  
Chong Shen ◽  
Weng Ho ◽  
Dongsheng Zhao
Keyword(s):  
Topological Space ◽  
Domain Theory ◽  
Scott Topology ◽  
Convergence Structure ◽  
Equivalent Conditions

In their attempt to develop domain theory in situ T0 spaces, Zhao and Ho introduced a new topology defined by irreducible sets of a resident topological space, called the SI-topology. Notably, the SI-topology of the Alexandroff topology of posets is exactly the Scott topology, and so the SI-topology can be seen as a generalisation of the Scott topology in the context of general T0 spaces. It is well known that the convergence structure that induces the Scott topology is the Scott-convergence - also known as lim-inf convergence by some authors. Till now, it is not known which convergence structure induces the SI-topology of a given T0 space. In this paper, we fill in this gap in the literature by providing a convergence structure, called the SI-convergence structure, that induces the SI-topology. Additionally, we introduce the notion of I-continuity that is closely related to the SI-convergence structure, but distinct from the existing notion of SI-continuity (introduced by Zhao and Ho earlier). For SI-continuity, we obtain here some equivalent conditions for it. Finally, we give some examples of non-Alexandroff SI-continuous spaces.

scholarly journals Continuity of Partially Ordered Soft Sets via Soft Scott Topology and Soft Sobrification

2014 ◽  
Vol 9 ◽  
pp. 79-88
Author(s):  
A.F. Sayed
Keyword(s):  
Computer Science ◽  
Theoretical Computer Science ◽  
Soft Set ◽  
Domain Theory ◽  
Scott Topology ◽  
Soft Sets ◽  
Theoretical Computer ◽  
Partially Ordered ◽  
Completely Distributive ◽  
Characterization Theorems

This paper, based on the concept of partially ordered soft sets (possets, for short) which proposed by Tanay and Yaylali [23], we will give some other concepts which are developing the possets and helped us in obtaining a generalization of some important results in domain theory which has an important and central role in theoretical computer science. Moreover, We will establish some characterization theorems for continuity of possets by the technique of embedded soft bases and soft sobrification via soft Scott topology, stressing soft order properties of the soft Scott topology of possets and rich interplay between topological and soft order-theoretical aspects of possets. We will see that continuous possets are all embedded soft bases for continuous directed completely partially ordered soft set (i.e., soft domains), and vice versa. Thus, one can then deduce properties of continuous possets directly from the properties of continuous soft domains by treating them as embedded bases for continuous soft domains. We will see also that a posset is continuous if its soft Scott topology is a complete completely distributive soft lattice.

scholarly journals Bicompleteness of the fine quasi-uniformity

1991 ◽  
Vol 109 (1) ◽  
pp. 167-186 ◽  
Author(s):  
Hans-Peter A. Künzi ◽  
Nathalie Ferrario
Keyword(s):  
Topological Space ◽  
Metrizable Space ◽  
Topological Spaces ◽  
Sober Space

AbstractA characterization of the topological spaces that possess a bicomplete fine quasi-uniformity is obtained. In particular we show that the fine quasi-uniformity of each sober space, of each first-countable T1-space and of each quasi-pseudo-metrizable space is bicomplete. Moreover we give examples of T1-spaces that do not admit a bicomplete quasi-uniformity.We obtain several conditions under which the semi-continuous quasi-uniformity of a topological space is bicomplete and observe that the well-monotone covering quasiuniformity of a topological space is bicomplete if and only if the space is quasi-sober.

scholarly journals Dcpo models of T1 spaces

10.29007/prcv ◽  
2018 ◽  
Author(s):  
Zhao Dongsheng ◽  
Xi Xiaoyong
Keyword(s):  
Topological Space ◽  
Positive Answer ◽  
Scott Topology ◽  
Maximal Points

A poset model of a topological space X is a poset P such that the subspace Max(P) of the Scott space ΣP consisting of all maximal points of P is homeomorphic to X. Every T<sub>1</sub> space has a (bounded complete algebraic) poset model. It is, however, not known whether every T<sub>1</sub> space has a dcpo model and whether every sober T<sub>1</sub> space has a dcpo model whose Scott topology is sober. In this paper we give a positive answer to these two problems. For each T<sub>1</sub> space X we shall construct a dcpo A that is a model of X, and prove that X is sober if and only if the Scott topology of A is sober. One useful by-product is a method that can be used to construct more non-sober dcpos.

scholarly journals RegularL-fuzzy topological spaces and their topological modifications

2000 ◽  
Vol 23 (10) ◽  
pp. 687-695 ◽  
Author(s):  
T. Kubiak ◽  
M. A. de Prada Vicente
Keyword(s):  
Topological Space ◽  
The Other ◽  
Regular Space ◽  
Continuous Lattice ◽  
Topological Spaces ◽  
Scott Topology ◽  
Fuzzy Topological Spaces

ForLa continuous lattice with its Scott topology, the functorιLmakes every regularL-topological space into a regular space and so does the functorωLthe other way around. This has previously been known to hold in the restrictive class of the so-called weakly induced spaces. The concepts ofH-Lindelöfness (á la Hutton compactness) is introduced and characterized in terms of certain filters. RegularH-Lindelöf spaces are shown to be normal.

Directed complete poset models of T1 spaces

2016 ◽  
Vol 164 (1) ◽  
pp. 125-134 ◽  
Author(s):  
DONGSHENG ZHAO ◽  
XIAOYONG XI
Keyword(s):  
Topological Space ◽  
Positive Answer ◽  
Scott Topology ◽  
Maximal Points

AbstractA poset model of a topological space X is a poset P such that the subspace Max(P) of the Scott space ΣP is homeomorphic to X, where Max(P) is the set of all maximal points of P. Every T1 space has a (bounded complete algebraic) poset model. It was, however, not known whether every T1 space has a directed complete poset model and whether every sober T1 space has a directed complete poset model whose Scott topology is sober. In this paper we give a positive answer to each of these two problems. For each T1 space X, we shall construct a directed complete poset E that is a model of X, and prove that X is sober if and only if the Scott space Σ E is sober. One useful by-product is a method for constructing more directed complete posets whose Scott topology is not sober.

Well-filtered spaces and their dcpo models

2015 ◽  
Vol 27 (4) ◽  
pp. 507-515 ◽  
Author(s):  
XIAOYONG XI ◽  
DONGSHENG ZHAO
Keyword(s):  
Topological Space ◽  
Negative Answer ◽  
Point Space ◽  
Open Set ◽  
Sober Space ◽  
Maximal Point ◽  
Saturated Sets

A topological space X is called well-filtered if for any filtered family $\mathcal{F}$ of compact saturated sets and an open set U, ∩ $\mathcal{F}$ ⊆ U implies F ⊆ U for some F ∈ $\mathcal{F}$. Every sober space is well-filtered and the converse is not true. A dcpo (directed complete poset) is called well-filtered if its Scott space is well-filtered. In 1991, Heckmann asked whether every UK-admitting (the same as well-filtered) dcpo is sober. In 2001, Kou constructed a counterexample to give a negative answer. In this paper, for each T1 space X we consider a dcpo D(X) whose maximal point space is homeomorphic to X and prove that X is well-filtered if and only if D(X) is well-filtered. The main result proved here enables us to construct new well-filtered dcpos that are not sober (only one such example is known by now). A space will be called K-closed if the intersection of every filtered family of compact saturated sets is compact. Every well-filtered space is K-closed. Some similar results on K-closed spaces are also proved.

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