AN APPROACH TO $\mathcal{M}$-SCOTT TOPOLOGY

2012 ◽  
Author(s):  
YUNPENG WEI ◽  
LINGXIA LU ◽  
WEI YAO
Keyword(s):  
Scott Topology

scholarly journals Coincidence of the upper Vietoris topology and the Scott topology

2021 ◽  
Vol 288 ◽  
pp. 107480 ◽  
Author(s):  
Xiaoquan Xu ◽  
Zhongqiang Yang
Keyword(s):  
Vietoris Topology ◽  
Scott Topology

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.

Spaces of maximal points

1997 ◽  
Vol 7 (5) ◽  
pp. 543-555 ◽  
Author(s):  
JIMMIE LAWSON
Keyword(s):  
Metric Spaces ◽  
Scott Topology ◽  
Maximal Points

This paper shows that it is precisely the complete metrizable separable metric spaces that can be realized as the set of maximal points of an ω-continuous dcpo, where the set of maximal points is topologized with the relative Scott topology.

AN ABSTRACT ELEMENTARY CLASS NONAXIOMATIZABLE IN

2019 ◽  
Vol 84 (3) ◽  
pp. 1240-1251
Author(s):  
SIMON HENRY
Keyword(s):  
Regular Cardinal ◽  
Vector Spaces ◽  
Left Adjoint ◽  
Scott Topology ◽  
Elementary Class ◽  
Closed Fields ◽  
Uncountable Cardinal ◽  
Image Position ◽  
Bounded Below ◽  
Abstract Elementary Class

AbstractWe show that for any uncountable cardinal λ, the category of sets of cardinality at least λ and monomorphisms between them cannot appear as the category of points of a topos, in particular is not the category of models of a ${L_{\infty ,\omega }}$-theory. More generally we show that for any regular cardinal $\kappa < \lambda$ it is neither the category of κ-points of a κ-topos, in particular, nor the category of models of a ${L_{\infty ,\kappa }}$-theory.The proof relies on the construction of a categorified version of the Scott topology, which constitute a left adjoint to the functor sending any topos to its category of points and the computation of this left adjoint evaluated on the category of sets of cardinality at least λ and monomorphisms between them. The same techniques also apply to a few other categories. At least to the category of vector spaces of with bounded below dimension and the category of algebraic closed fields of fixed characteristic with bounded below transcendence degree.

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.

Fixed points of contractive maps on dcpo's

2013 ◽  
Vol 24 (1) ◽  
Author(s):  
E. COLEBUNDERS ◽  
S. DE WACHTER ◽  
R. LOWEN
Keyword(s):  
Fixed Point ◽  
Fixed Points ◽  
Fixed Point Theorems ◽  
Scott Topology ◽  
Study Approach ◽  
Special Cases ◽  
Monotone Maps ◽  
Contractive Maps ◽  
Approach Spaces

In this paper we study approach structures on dcpo's. A dcpo (X, ≤) will be endowed with several other structures: the Scott topology; an approach structure generated by a collection of weightable quasi metrics onX; and a collectionof weights corresponding to the quasi metrics. Understanding the interaction between these structures onXwill eventually lead to some fixed-point theorems for the morphisms in the category of approach spaces, which are called contractions. Existing fixed-point theorems on both monotone and non-monotone maps are obtained as special cases.

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.

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