The Tukey Order and Subsets of ω 1

Gartside, Paul; Mamatelashvili, Ana

doi:10.1007/s11083-017-9423-6

Cited by 7 publications

(5 citation statements)

References 21 publications

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“…'Calibre (κ, λ, λ)' is abbreviated to 'calibre (κ, λ)' and 'calibre (κ, κ)' is abbreviated to 'calibre κ'. The following two straightforward facts were proven in [9] and [10], respectively.…”

Section: General Calibresmentioning

confidence: 81%

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Tukey Order, Calibres and the Rationals

Gartside¹,

Mamatelashvili²

2016

Preprint

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One partially ordered set, Q, is a Tukey quotient of another, Pdenoted P ≥ T Q -if there is a map φ : P → Q carrying cofinal sets of P to cofinal sets of Q. Let X be a space and denote by K(X) the set of compact subsets of X, ordered by inclusion. For certain separable metrizable spaces M , Tukey upper and lower bounds of K(M ) are calculated. Results on invariants of K(M )'s are deduced. The structure of all K(M )'s under ≤ T is investigated. Particular emphasis is placed on the position of K(M ) when M is: completely metrizable, the rationals Q, co-analytic or analytic.

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Section: General Calibresmentioning

confidence: 81%

“…Applications were given to function spaces and to certain compact spaces (Gul'ko compacta) arising in functional analysis. (The authors have also, see [10], examined the Tukey structure of K(S)'s where S is a subspace of ω 1 . )…”

Section: Introductionmentioning

confidence: 99%

Tukey Order, Calibres and the Rationals

Gartside¹,

Mamatelashvili²

2016

Preprint

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“…Using van Douwen's results, Nickolas and Tkachenko [16,17] computed the character of the free topological groups F (X) and free topological abelian groups A(X) over the spaces X considered by van Douwen. Independently of the present work, and concurrently, Gartside and Mamatelashvili [11,12,13] considered the cofinalities and the cofinal structure of the partially ordered sets of compact sets for various spaces. For non-scattered totally imperfect separable metric spaces, and for complete metric spaces of uncountable weight less than ℵ ω , they identified the exact cofinal structures of families of compact sets.…”

Section: Introduction and Related Workmentioning

confidence: 95%

A classification of the cofinal structures of precompacta

Eshed

Ferrer

Hernández

et al. 2020

Annals of Pure and Applied Logic

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We provide a complete classification of the possible cofinal structures of the families of precompact (totally bounded) sets in general metric spaces, and compact sets in general complete metric spaces. Using this classification, we classify the cofinal structure of local bases in the groups C(X, R) of continuous real-valued functions on complete metric spaces X, with respect to the compact-open topology.

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“…if there is a map ϕ : A → C so that whenever F ⊆ A is cofinal relative to B, then ϕ[F] is cofinal relative to D. This definition is inspired by Paul Gartside's work on the Tukey order [5].…”

Section: Introductionmentioning

confidence: 99%

Selection Games on Continuous Functions

Caruvana,

Holshouser

2019

Preprint

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In this paper we study the selection principle of closed discrete selection, first researched by Tkachuk in [13] and strengthened by Clontz, Holshouser in [3], in set-open topologies on the space of continuous real-valued functions. Adapting the techniques involving point-picking games on X and C p (X), the current authors showed similar equivalences in [1] involving the compact subsets of X and C k (X). By pursuing a bitopological setting, we have touched upon a unifying framework which involves three basic techniques: general game duality via reflections (Clontz), general game equivalence via topological connections, and strengthening of strategies (Pawlikowski and Tkachuk). Moreover, we develop a framework which identifies topological notions to match with generalized versions of the point-open game. Definitions and PreliminariesDefinition 1. Let X be a space and A ⊆ ℘(X). We say that A is an ideal-base if, for A 1 , A 2 ∈ A, there existsDefinition 2. For a topological space X and a collection A ⊆ ℘(X), we let Ā = {cl X (A) : A ∈ A}.Definition 3. Fix a topological space X and a collection A ⊆ ℘(X). Then• we let C p (X) denote the set of all continuous functions X → R endowed with the topology of point-wise convergence; we also let 0 be the function which identically zero.• we let C k (X) denote the set of all continuous functions X → R endowed with the topology of uniform convergence on compact subsets of X; we will write• in general, we let C A (X) denote the set of all continuous functions X → R endowed with the A-open topology; we will writeNotice that, for the sets of the form [f ; A, ε] to be a base for the topology C A (X), then A must be an ideal-base.Definition 4. For a topological space X, we let K(X) denote the family of all non-empty compact subsets of X.Definition 5. Let X be a topological space. We say that A ⊆ X is R-bounded if, for every continuous f :In this paper, we will be concerned with selection principles and related games. For classical results, basic tools, and notation, the authors recommend [10] and [7]. Definition 6. Consider collections A and B and an ordinal α. The corresponding selection principles are defined as follows:• S α fin (A, B) is the assertion that, given any {A ξ : ξ ∈ α} ⊆ A, there exists {F ξ : ξ ∈ α} so that, for each ξ ∈ α, F ξ is a finite subset of A ξ (denoted as F ξ ∈ [A ξ ] <ω hereinafter) and {F ξ : ξ ∈ α} ∈ B, and• S α 1 (A, B) is the assertion that, given any {A ξ : ξ ∈ α} ⊆ A, there exists {x ξ : ξ ∈ α} so that, for each ξ ∈ α, x ξ ∈ A ξ and {x ξ : ξ ∈ α} ∈ B.We suppress the superscript when α = ω; i.e., S 1 (A, B) = S ω 1 (A, B).

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The Tukey Order and Subsets of ω 1

Cited by 7 publications

References 21 publications

Tukey Order, Calibres and the Rationals

Tukey Order, Calibres and the Rationals

A classification of the cofinal structures of precompacta

Selection Games on Continuous Functions

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