The Tukey Order on Compact Subsets of Separable Metric Spaces

Gartside, Paul; Mamatelashvili, Ana

doi:10.1017/jsl.2015.49

Cited by 14 publications

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“…We completely solve the problem of which sets S ⊆ ω 1 have the property that K(M ) ≥ T K(S) for some separable metrizable M . Recently, in [12], the authors constructed 2 c -many Tukey classes of K(M )'s, with M separable and metrizable, and used them to show that there are large families of function spaces corresponding to these M 's. Here we use Todorčević's 2 ω1 -many distinct Tukey classes of K(S)'s, where S is a subset of ω 1 , to derive similar results about function spaces.…”

Section: Introductionmentioning

confidence: 90%

“…In this section we will give some basic connections of the (relative) Tukey ordering with order properties of posets. Sketches of proofs of some of these results were presented in [12] and we omit them here. Full proofs can be found in [16].…”

Section: Definitions and Basic Propertiesmentioning

confidence: 99%

“…separable and completely metrizable). Posets of the form K(M ), where M is separable and metrizable have been studied extensively [5,9,12]. In works of Cascales, Orihuela and Tkachuk a special interest has been given to investigating properties of spaces X such that K(M ) ≥ T K(X) for some separable metrizable M [3,4].…”

Section: Introductionmentioning

confidence: 99%

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The Tukey Order and Subsets of ω 1

Gartside

Mamatelashvili

2017

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One partially ordered set, Q, is a Tukey quotient of another, P , if there is a map φ : P → Q carrying cofinal sets of P to cofinal sets of Q. Two partial orders which are mutual Tukey quotients are said to be Tukey equivalent. Let X be a space and denote by K(X) the set of compact subsets of X, ordered by inclusion. The principal object of this paper is to analyze the Tukey equivalence classes of K(S) corresponding to various subspaces S of ω 1 , their Tukey invariants, and hence the Tukey relations between them. It is shown that ω ω is a strict Tukey quotient of Σ(ω ω 1 ) and thus we distinguish between two Tukey classes out of Isbell's ten partially ordered sets from [14]. The relationships between Tukey equivalence classes of K(S), where S is a subspace of ω 1 , and K(M ), where M is a separable metrizable space, are revealed. Applications are given to function spaces.2010 Mathematics Subject Classification. 03E04, 06A07, 54F05 (54C35, 54E35).

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Section: Introductionmentioning

confidence: 90%

Section: Definitions and Basic Propertiesmentioning

confidence: 99%

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The Tukey Order and Subsets of ω 1

Gartside

Mamatelashvili

2017

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“…The next two results are from a paper [5] by the second author and Mamatelashvili. The first says when subsets of K(M) have upper bounds, it implies that K(M) is countably directed.…”

Section: Directed Setsmentioning

confidence: 95%

“…If P ′ = P , then we may abbreviate this to P ≥ T (Q ′ , Q). See [5] for the proof of the next lemma. The non-relative version is well-known.…”

Section: Directed Setsmentioning

confidence: 99%

P-paracompact and P-metrizable spaces

Feng

Gartside

Morgan

2015

Topology and its Applications

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Available online xxxx MSC: 54A05 54D70 54E35Let P be a directed set and X a space. A collection C of subsets of X is P -locally finite if C = {C p : p ∈ P } where (i) if p ≤ p then C p ⊆ C p and (ii) each C p is locally finite. Then X is P -paracompact if every open cover has a P -locally finite open refinement. Further, X is P -metrizable if it has a (P × N)-locally finite base. This work provides the first detailed study of P -paracompact and P -metrizable spaces, particularly in the case when P is a K(M ), the set of all compact subsets of a separable metrizable space M ordered by set inclusion. Z. Feng et al. / Topology and its Applications 191 (2015) 97-118 is (i) N-metacompact, (ii) K(ω ω )-metacompact, or (iii) K(M )-metacompact, for some separable metrizable space M (respectively). The equivalence of (i) and (i) is due to Gruenhage [7], while the other two equivalences are due to Garcia, Orihuela and Oncina [3]. The first two authors gave in [2] a systematic and uniform development of the theory of those compact spaces X such that X 2 \ Δ is P -metacompact (P -Eberlein compact), giving alternative characterizations in terms of almost subbases, bases, networks and point networks.A key result from [2] is that a P -Eberlein compact space is Corson compact (embeds in a Σ-product of lines) if P has calibre (ω 1 , ω) (every uncountable subset of P contains an infinite subset with an upper bound), but if P is not calibre (ω 1 , ω) then every compact space of weight no more than ω 1 is P -Eberlein compact. This demonstrates the critical role that calibre (ω 1 , ω) plays. However Gul'ko (and so Talagrand and Eberlein) compacta have pleasant properties that general Corson compacta do not (for example, ccc Gul'ko compacta are metrizable in ZFC). The proofs use special properties of directed sets of the form K(M ). The authors would like to know what -if anything -is special about directed sets of the form K(M ) as compared to general directed sets with calibre (ω 1 , ω).Here we show (Theorems 47 and 49) that a pseudocompact space X such that X 2 \ Δ is P -paracompact for some P with calibre (ω 1 , ω) is metrizable. However, if P is not calibre (ω 1 , ω) then there is a compact non-metrizable space X such that X 2 \Δ is P -paracompact. This gives an 'optimal' extension of Gruenhage's result [7], that a compact space X with paracompact X 2 \ Δ is metrizable. We also show that a separable space is Lindelöf if P -paracompact, and metrizable if P -metrizable, for some P with calibre (ω 1 , ω).However developing further results about P -paracompact and P -metrizable spaces apparently needs P to be a K(M ). These results depend on alternative characterizations which do not (overtly) refer to a directed set. For example, we show (Theorem 50) that a space is K(M )-metrizable for some separable metrizable M if and only if it has a weakly σ-locally finite base (a base B = n B n where for each point x we have B = {B s : x is locally finite in B s }). With these in place we show that a first countable space is paracompact if ...

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Directed sets of topology: Tukey representation and rejection

Feng,

Gartside

2024

Rev. Real Acad. Cienc. Exactas Fis. Nat. Ser. A-Mat.

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The Tukey Order on Compact Subsets of Separable Metric Spaces

Cited by 14 publications

References 20 publications

The Tukey Order and Subsets of ω 1

The Tukey Order and Subsets of ω 1

P-paracompact and P-metrizable spaces

Directed sets of topology: Tukey representation and rejection

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