Topological aspects of Poset spaces

Mummert, Carl; Stephan, Frank

doi:10.1307/mmj/1272376025

Cited by 11 publications

(8 citation statements)

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“…Stephan [23] have shown that the spaces that are homeomorphic to the maximal elements of some ω-continuous domain are precisely the countably based strong Choquet spaces that satisfy the T 1 -separation axiom (K. Martin had previously shown that the maximal elements are strong Choquet).…”

Section: ⊓ ⊔mentioning

confidence: 99%

Quasi-Polish spaces

Brecht

2013

Annals of Pure and Applied Logic

124

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In the presence of suitable power spaces, compactness of X can be characterized as the singleton {X} being open in the space O(X) of open subsets of X. Equivalently, this means that universal quantification over a compact space preserves open predicates.Using the language of represented spaces, one can make sense of notions such as a Σ 0 2subset of the space of Σ 0 2 -subsets of a given space. This suggests higher-order analogues to compactness: We can, e.g. , investigate the spaces X where {X} is a ∆ 0 2 -subset of the space of ∆ 0 2 -subsets of X. Call this notion ∇-compactness. As ∆ 0 2 is self-dual, we find that both universal and existential quantifier over ∇-compact spaces preserve ∆ 0 2 predicates. Recall that a space is called Noetherian iff every subset is compact. Within the setting of Quasi-Polish spaces, we can fully characterize the ∇-compact spaces: A Quasi-Polish space is Noetherian iff it is ∇-compact. Note that the restriction to Quasi-Polish spaces is sufficiently general to include plenty of examples.

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Section: ⊓ ⊔mentioning

confidence: 99%

Quasi-Polish spaces

Brecht

2013

Annals of Pure and Applied Logic

124

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“…(3) The maximal antichain space A co max is not quasi-Polish. For (1), [43,Theorem 6.1] by Mummert and Stephan implies that the Gandy-Harrington space can be represented as the maximal elements of an omega-algebraic domain. Therefore, the Gandy-Harrington space embeds as a (necessarily strict) co-analytic subset of a quasi-Polish space.…”

Section: Example 331 (Mccarthymentioning

confidence: 99%

Enumeration degrees and non-metrizable topology

Kihara,

Ng,

Pauly

2019

Preprint

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The enumeration degrees of sets of natural numbers can be identified with the degrees of difficulty of enumerating neighborhood bases of points in a universal second-countable T 0 -space (e.g. the ω-power of the Sierpiński space). Hence, every represented second-countable T 0 -space determines a collection of enumeration degrees. For instance, Cantor space captures the total degrees, and the Hilbert cube captures the continuous degrees by definition. Based on these observations, we utilize general topology (particularly non-metrizable topology) to establish a classification theory of enumeration degrees of sets of natural numbers. ContentsTAKAYUKI KIHARA, KENG MENG NG, AND ARNO PAULY 5.3. Degrees of points: T 2.5 -topology 45 5.4. Degrees of points: submetrizable topology 50 5.5. Degrees of points: G δ -topology 56 5.6. Quasi-Polish topology 59 6. Cs-networks and non-second-countability 62 6.1. Regular-like networks and closure representation 63 6.2. Near quasi-minimality 66 6.3. Borel extension topology 67 7. Proofs for Section 4 69 7.1. T 0 -degrees which are not T 1 69 7.2. T 1 -degrees which are not T 2 84 7.3. T 2 -degrees which are not T 2.5 93 7.4. T 2.5 -degrees which are not T 3 97 8. Open Questions 99 References 101Recall that a subset of ω is d-c.e. if it is the difference of two c.e. sets, and co-d-c.e. if it is the complement of a d-c.e. set, that is, the union A ∪ P of a c.e. set P and co-c.e. set A such that A and P are disjoint. Note that an enumeration degree contains a co-d-c.e. set if and only if it contains a 3-c.e. set. Definition 3.5. Let a be an enumeration degree.(1) We say that a is co-d-CEA if a contains a set of the form (X ⊕ X c ) ⊕ (A ∪ P ) for some X, A, P ⊆ ω such that P and A c are X-c.e., and A and P are disjoint. (2) Generally, we say that a is Γ-above if a contains a set of the form (X ⊕ X c ) ⊕ Z such that Z is Γ in X.(3) We say that a is doubled co-d-CEA if a contains a set of the form (X ⊕ X c ) ⊕ (A ∪ P ) ⊕ (B ∪ N) for some X, A, B, P, N ⊆ ω such that P , N, and A ∪ B c are X-c.e., and that A, B, P and N are pairwise disjoint.

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“…(f) Domain theory and associated topologies are studied in RM ( [64,78]), and nets take central stage in domain theory in [44,45,47]. (g) Nets are used in topological dynamics ( [41]), which is studied in the proof mining program ( [43,60]).…”

Section: Aim and Motivationmentioning

confidence: 99%

Nets and Reverse Mathematics, a pilot study

Sanders

2019

Preprint

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Nets are generalisations of sequences involving possibly uncountable index sets; this notion was introduced about a century ago by Moore and Smith. They also established the generalisation to nets of various basic theorems of analysis due to Bolzano-Weierstrass, Dini, Arzelà, and others. More recently, nets are central to the development of domain theory, providing intuitive definitions of the associated Scott and Lawson topologies, among others. This paper deals with the Reverse Mathematics study of basic theorems about nets. We restrict ourselves to nets indexed by subsets of Baire space, and therefore third-order arithmetic, as such nets suffice to obtain our main results. Over Kohlenbach's base theory of higher-order Reverse Mathematics, the Bolzano-Weierstrass theorem for nets implies the Heine-Borel theorem for uncountable covers. We establish similar results for other basic theorems about nets and even some equivalences, e.g. for Dini's theorem for nets. Finally, we show that replacing nets by sequences is hard, but that replacing sequences by nets can obviate the need for the Axiom of Choice, a foundational concern in domain theory. In an appendix, we study the power of more general index sets, establishing that the 'size' of a net is directly proportional to the power of the associated convergence theorem.

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Topological aspects of Poset spaces

Cited by 11 publications

References 10 publications

Quasi-Polish spaces

Quasi-Polish spaces

Enumeration degrees and non-metrizable topology

Nets and Reverse Mathematics, a pilot study

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