Some further results on order-convergence in posets

Wang, Kaiyun; Zhao, Bin

doi:10.1016/j.topol.2012.09.018

Cited by 14 publications

(9 citation statements)

References 8 publications

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“…To this end, one needs to observe that in L every (nontrivial) monotonic net has to either be eventually in L k (and therefore converges to 0 k ) for some k or be eventually in one of the sets {a k It is well-known that order convergence fails to be topological, in general; not even when P is a complete lattice. Characterizations of posets having the property that O 2 -order convergence is topological can be found in [25,24,23]. For lattices, other characterizations have been thoroughly studied (in the language of filters) in [15,13,9,10].…”

Section: Introductionmentioning

confidence: 99%

On different modes of order convergence and some applications

Abela¹,

Chetcuti²,

Weber³

2020

Preprint

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Different notions for order convergence have been considered by various authors. Associated to every notion of order convergence corresponds a topology, defined by taking as the closed sets those subsets of the poset satisfying that no net in them order converges to a point that is outside of the set. We shall give a thorough overview of these different notions and provide a systematic comparison of the associated topologies. Then, in the last section we shall give an application of this study by giving a result on von Neumann algebras complementing the study started in [8]. We show that for every atomic von Neumann algebra (not necessarily σ-finite) the restriction of the order topology to bounded parts of M coincides with the restriction of the σ-strong topology s(M, M * ). We recall that the methods of [8] rest heavily on the assumption of σ-finiteness. Further to this, for a semi-finite measure space, we shall give a complete picture of the relations between the topologies on L ∞ associated with the duality L 1 , L ∞ and its order topology.

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Section: Introductionmentioning

confidence: 99%

On different modes of order convergence and some applications

Abela¹,

Chetcuti²,

Weber³

2020

Preprint

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“…Many convergent classes in posets were studied in [3,[8][9][10][11][12]. By di erent convergences, not only many notions of continuity are characterized, but also they make order and topology across each other.…”

Section: Introductionmentioning

confidence: 99%

On a new convergence in topological spaces

Ruan

2019

Open Mathematics

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In this paper, we introduce a new way-below relation in T0 topological spaces based on cuts and give the concepts of SI2-continuous spaces and weakly irreducible topologies. It is proved that a space is SI2-continuous if and only if its weakly irreducible topology is completely distributive under inclusion order. Finally, we introduce the concept of 𝓓-convergence and show that a space is SI2-continuous if and only if its 𝓓-convergence with respect to the topology τSI2(X) is topological. In general, a space is SI-continuous if and only if its 𝓓-convergence with respect to the topology τSI(X) is topological.

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“…Hence, much works has been done to characterize those special posets in which the O-convergence is topological. The most recent result in [13] shows that the O-convergence in a poset which satisfies condition ( ) is topological if and only if the poset is O-doubly continuous. This means that for a special class of posets, a sufficient and necessary condition for O-convergence being topological is obtained.…”

Section: Introductionmentioning

confidence: 99%

O1-convergence in partially ordered sets

Sun¹,

Li²,

Fan³

2019

J. Nonlinear Sci. Appl.

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Based on the introduction of notions of S * -doubly continuous posets and B-topology in [T. Sun, Q. G. Li, L. K. Guo, Topology Appl., 207 (2016), 156-166], in this paper, we further propose the concept of B-consistent S * -doubly continuous posets and prove that the O 1 -convergence in a poset is topological if and only if the poset is a B-consistent S * -doubly continuous poset. This is the main result which can be seen as a sufficient and necessary condition for the O 1 -convergence in a poset being topological. Additionally, in order to present natural examples of posets which satisfy such condition, several special sub-classes of B-consistent S * -doubly continuous posets are investigated.

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Some further results on order-convergence in posets

Cited by 14 publications

References 8 publications

On different modes of order convergence and some applications

On different modes of order convergence and some applications

On a new convergence in topological spaces

O1-convergence in partially ordered sets

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