Domain Representations Induced by Dyadic Subbases

Tsuiki, Hideki; Tsukamoyo, Yasuyuki

doi:10.2168/lmcs-11(1:17)2015

Cited by 3 publications

(11 citation statements)

References 12 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The set D S is an algebraic pointed dcpo that is the ideal completion of K S . We quote some results known from [6,7]: If S is a proper dyadic subbase of a regular Hausdorff space X, then X is embedded in the space of minimal elements of D S \ K S . Moreover, if X is compact in addition, then we have a quotient map ρ S : D S \ K S → X and X is homeomorphic to the space of minimal elements of D S \ K S .…”

Section: 2mentioning

confidence: 99%

“…Every finite prefix of the bottomed sequence ϕ S (x) can be considered as a finite time state of the output of computation, and the properties of the set K S := {ϕ S (x)| n : x ∈ X, n < ω} of these sequences have been studied. If S is proper and K S is a conditional upper semilattice with least element (cusl), then we obtain an admissible domain representation of X [6]. Whether K S is a cusl depends not only on S itself but also on the enumeration of S. It has been proved that K S is a cusl regardless of the enumeration of S if and only if S is strongly proper [7].…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Existence of strongly proper dyadic subbases

Tsukamoto¹

2017

Logical Methods in Computer Science ; Volume 13

View full text Add to dashboard Cite

We consider a topological space with its subbase which induces a coding for each point. Every second-countable Hausdorff space has a subbase that is the union of countably many pairs of disjoint open subsets. A dyadic subbase is such a subbase with a fixed enumeration. If a dyadic subbase is given, then we obtain a domain representation of the given space. The properness and the strong properness of dyadic subbases have been studied, and it is known that every strongly proper dyadic subbase induces an admissible domain representation regardless of its enumeration. We show that every locally compact separable metric space has a strongly proper dyadic subbase.

show abstract

Section: 2mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Existence of strongly proper dyadic subbases

Tsukamoto¹

2017

Logical Methods in Computer Science ; Volume 13

View full text Add to dashboard Cite

show abstract

“…This section briefly reviews proper dyadic subbases (Tsuiki 2004b;Tsuiki and Tsukamoto 2015). Throughout this section, X = (X, O) is a second-countable Hausdorff space.…”

Section: Dyadic Subbases Of Topological Spacesmentioning

confidence: 99%

“…We define the minimal limit set M S of D S as the set of minimal elements of L S . We have the following (Tsuiki and Tsukamoto 2015).…”

Section: Dyadic Subbases Of Topological Spacesmentioning

confidence: 99%

See 1 more Smart Citation

Properties of domain representations of spaces through dyadic subbases

Tsukamoto¹,

Tsuiki²

2016

Math. Struct. Comp. Sci.

Self Cite

View full text Add to dashboard Cite

A dyadic subbase S of a topological space X is a subbase consisting of a countable collection of pairs of open subsets that are exteriors of each other. If a dyadic subbase S is proper, then we can construct a dcpo DS in which X is embedded. We study properties of S with respect to two aspects. (i) Whether the dcpo DS is consistently complete depends on not only S itself but also the enumeration of S. We give a characterization of S that induces the consistent completeness of DS regardless of its enumeration. (ii) If the space X is regular Hausdorff, then X is embedded in the minimal limit set of DS. We construct an example of a Hausdorff but non-regular space with a dyadic subbase S such that the minimal limit set of DS is empty.

show abstract

Computable dyadic subbases and $\mathbf{T}^ω$-representations of compact sets

Pauly¹,

Tsuiki²

2016

Preprint

Self Cite

View full text Add to dashboard Cite

We explore representing the compact subsets of a given represented space by infinite sequences over Plotkin's T. We show that computably compact computable metric spaces admit representations of their compact subsets in such a way that compact sets are essentially underspecified points. We can even ensure that a name of an n-element compact set contains n occurrences of ⊥. We undergo this study effectively and show that such a T ωrepresentation is effectively obtained from structures of computably compact computable metric spaces. As an application, we prove some statements about the Weihrauch degree of closed choice for finite subsets of computably compact computable metric spaces.Along the way, we introduce the notion of a computable dyadic subbase, and prove that every computably compact computable metric space admits a proper computable dyadic subbase.

show abstract

Domain Representations Induced by Dyadic Subbases

Cited by 3 publications

References 12 publications

Existence of strongly proper dyadic subbases

Existence of strongly proper dyadic subbases

Properties of domain representations of spaces through dyadic subbases

Computable dyadic subbases and $\mathbf{T}^ω$-representations of compact sets

Contact Info

Product

Resources

About