Two topologies on the lattice of Scott closed subsets

Chen, Yu; Kou, Hui; Lyu, Zhenchao

doi:10.1016/j.topol.2021.107918

Cited by 6 publications

(4 citation statements)

References 12 publications

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“…In domain theory, there is a correspondence between the continuity of a dcpo L and the continuity of the lattice of its Scott open subsets σ(L). More surprisingly, the continuity of a dcpo L and the continuity of its lattice of its Scott closed subsets Γ(L) coincide [18]. These correspondences can be extended to directed spaces very smoothly.…”

Section: Introductionmentioning

confidence: 78%

“…In [18], Chen and Kou give a uniform proof for the correspondence between the continuity of a dcpo and the lattice of its Scott closed subsets. Dcpos can be viewed as special directed spaces.…”

Section: The Lattice Of Closed Subsets Of Directed Spacesmentioning

confidence: 99%

“…Proposition 6.1. [18] Let L be a continuous lattice, if L satisfies the condition that υ(L op ) = σ(L op ), then (L op , σ(L op )) is a sober and locally compact space. Proposition 6.2.…”

Section: The Lattice Of Closed Subsets Of Directed Spacesmentioning

confidence: 99%

“…We have that (C(X), σ(C(X))) is sober and locally compact by Proposition 6.1. ✷ In [18], an adjunction between σ(P ) and σ(C(P )) serves as a useful tool in studying the relation between P and C(P ). It can be extended to directed spaces as well.…”

Section: D(mentioning

confidence: 99%

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Continuity on directed spaces

Chen¹,

Kou²,

Lyu³

2022

Preprint

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We first investigate two approximation relations on a T 0 topological space, the n-approximation and the d-approximation, which are generalizations of the waybelow relation on a dcpo. Different kinds of continuous spaces are defined by the two approximations and are all shown to be directed spaces. We will show that the continuity of a directed space is very similar to the continuity of a dcpo in many aspects, which indicates that the notion of directed spaces is a suitable topological extension of dcpos.The main results are: (1) A topological space is continuous iff it is a retract of an algebraic space.(2) a directed space X is core compact iff for any directed space Y , X × Y = X ⊗ Y , where X × Y and X ⊗ Y are the topological product and categorical product in DTop of X and Y respectively; (3) a directed space is continuous (resp., algebraic, quasicontinuous, quisialgebraic) iff the lattice of its closed subsets is continuous (resp., algebraic, quasicontinuous, quisialgebraic).

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Section: Introductionmentioning

confidence: 78%

Section: The Lattice Of Closed Subsets Of Directed Spacesmentioning

confidence: 99%

Section: The Lattice Of Closed Subsets Of Directed Spacesmentioning

confidence: 99%

Section: D(mentioning

confidence: 99%

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Continuity on directed spaces

Chen¹,

Kou²,

Lyu³

2022

Preprint

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Topological properties of the binary supremum function

Hertling

2022

Semigroup Forum

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We consider the binary supremum function $$\sup :Z\times Z\rightarrow Z$$ sup : Z × Z → Z on a sup semilattice Z and its topological properties with respect to the Scott topology and the product topology. It is well known that this function is continuous with respect to the Scott topology on $$Z\times Z$$ Z × Z . We show that it is open as well. Isbell has constructed several examples of complete lattices Z such that the binary supremum function on Z is discontinuous with respect to the product topology. Of course, in these cases the Scott topology on $$Z\times Z$$ Z × Z is strictly finer than the product topology. This raises the question whether there exists a complete lattice Z such that the Scott topology on $$Z\times Z$$ Z × Z is strictly finer than the product topology and such that the binary supremum function is continuous even with respect to the product topology. We construct such a lattice. Finally, by using any of the examples constructed by Isbell, we show the following result: Bounded completeness of a complete lattice Z is in general not inherited by the dcpo C(X, Z) of continuous functions from X to Z where X is a topological space and where on Z the Scott topology is considered. On the other hand, we show that bounded completeness of Z is inherited by C(X, Z) if the topology on X is the Scott topology of a partial order.

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H-closedness and absolute H-closedness

Chu,

2024

Semigroup Forum

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Two topologies on the lattice of Scott closed subsets

Cited by 6 publications

References 12 publications

Continuity on directed spaces

Continuity on directed spaces

Topological properties of the binary supremum function

H-closedness and absolute H-closedness

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