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.

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 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.

scholarly journals A Functional Analytic Description of Normal Spaces

1972 ◽  
Vol 24 (1) ◽  
pp. 45-49 ◽  
Author(s):  
E. Binz ◽  
W. Feldman
Keyword(s):  
Topological Space ◽  
Continuous Functions ◽  
Closed Ideal ◽  
Analytic Description ◽  
Completely Regular ◽  
Hausdorff Topological Space ◽  
Continuous Convergence ◽  
Convergence Structure

Throughout the paper, X will denote a completely regular (Hausdorff) topological space and C(X) the R-algebra of all real-valued continuous functions on X. When this algebra carries the continuous convergence structure [1], we write CC(X). We note that CC(X)is a complete [5] convergence R-algebra [1].Our description of normality reads as follows. A completely regular topological space X is normal if and only if CC(X)/J (endowed with the obvious quotient structure; see § 1) is complete for every closed ideal J ⊂ CC(X).

The Spatial Dimensions of Social Networks

2020 ◽  
pp. 367-383
Author(s):  
Zachary P. Neal
Keyword(s):  
Social Networks ◽  
Topological Space ◽  
Network Science ◽  
Spatial Dimensions

The first law of geography holds that everything is related to everything else, but near things are more related than distant things, where distance refers to topographical space. If a first law of network science exists, it would similarly hold that everything is related to everything else, but near things are more related than distant things, but where distance refers to topological space. Frequently these two laws collide, together holding that everything is related to everything else, but topographically and topologically near things are more related than topographically and topologically distant things. The focus of the spatial study of social networks lies in exploring a series of questions embedded in this combined law of geography and networks. This chapter explores the questions that have been asked and the answers that have been offered at the intersection of geography and networks.

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