Continuous images of closed sets in generalized Baire spaces

Lücke, Philipp; Schlicht, Philipp

doi:10.1007/s11856-015-1224-2

Cited by 15 publications

(51 citation statements)

References 17 publications

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“…Further background . It is consistent that the club filter on κ is not a continuous injective image of any closed subset of

κ^{κ}

(Lücke, Schlicht; ).…”

Section: The List Of Open Questionsmentioning

confidence: 94%

“…Every closed, non‐empty subset of

ω^{ω}

is a continuous image of the Baire space

ω^{ω}

(in fact it is even a retract of the whole space) [, Proposition 2.8]. Every closed subset of

κ^{κ}

can be written as [ T ] for some tree

T \subseteq κ^{< κ}

, however, by results from [, Theorem 1.5] there is always a tree T such that [ T ] is not a continuous image of

κ^{κ}

. Therefore it is interesting to ask whether the closed sets induced by trees with certain special properties (e.g.…”

Section: The List Of Open Questionsmentioning

confidence: 99%

“…Questions to questions refer to , where various sub‐classes of

Σ_{1}^{1}

‐sets are examined. Question Is it consistent that the club filter on κ is an injective continuous image of

κ^{κ}

?…”

Section: The List Of Open Questionsmentioning

confidence: 99%

“…Further background . This question is motivated by [, Proposition 1.13], which shows that

{bold-italicΣ}_{1}^{1} = S_{1}^{boldL}

in models of the form

L [x]

, with

x \subseteq κ

.…”

Section: The List Of Open Questionsmentioning

confidence: 99%

“…Question 3.21 (Friedman,Hyttinen,Kulikov;[22,37]) Is it consistent that 1 1 = Borel * ? Questions 3.22 to 3.27 questions refer to [49], where various sub-classes of 1 1 -sets are examined.…”

Section: Topology and Silver Dichotomymentioning

confidence: 99%

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Questions on generalised Baire spaces

Khomskii

Laguzzi

Löwe

et al. 2016

Mathematical Logic Qtrly

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show abstract

“…Further background . It is consistent that the club filter on κ is not a continuous injective image of any closed subset of

κ^{κ}

(Lücke, Schlicht; ).…”

Section: The List Of Open Questionsmentioning

confidence: 94%

“…Every closed, non‐empty subset of

ω^{ω}

is a continuous image of the Baire space

ω^{ω}

(in fact it is even a retract of the whole space) [, Proposition 2.8]. Every closed subset of

κ^{κ}

can be written as [ T ] for some tree

T \subseteq κ^{< κ}

, however, by results from [, Theorem 1.5] there is always a tree T such that [ T ] is not a continuous image of

κ^{κ}

. Therefore it is interesting to ask whether the closed sets induced by trees with certain special properties (e.g.…”

Section: The List Of Open Questionsmentioning

confidence: 99%

“…Questions to questions refer to , where various sub‐classes of

Σ_{1}^{1}

‐sets are examined. Question Is it consistent that the club filter on κ is an injective continuous image of

κ^{κ}

?…”

Section: The List Of Open Questionsmentioning

confidence: 99%

“…Further background . This question is motivated by [, Proposition 1.13], which shows that

{bold-italicΣ}_{1}^{1} = S_{1}^{boldL}

in models of the form

L [x]

, with

x \subseteq κ

.…”

Section: The List Of Open Questionsmentioning

confidence: 99%

Section: Topology and Silver Dichotomymentioning

confidence: 99%

See 3 more Smart Citations

Questions on generalised Baire spaces

Khomskii

Laguzzi

Löwe

et al. 2016

Mathematical Logic Qtrly

View full text Add to dashboard Cite

show abstract

The Hurewicz dichotomy for generalized Baire spaces

Luecke

Ros

Schlicht³

2016

Isr. J. Math.

Self Cite

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Abstract. By classical results of Hurewicz, Kechris and Saint-Raymond, an analytic subset of a Polish space X is covered by a Kσ subset of X if and only if it does not contain a closed-in-X subset homeomorphic to the Baire space ω ω. We consider the analogous statement (which we call Hurewicz dichotomy) for Σ 1 1 subsets of the generalized Baire space κ κ for a given uncountable cardinal κ with κ = κ <κ . We show that the statement that this dichotomy holds at all uncountable regular cardinals is consistent with the axioms of ZFC together with GCH and large cardinal axioms. In contrast, we show that the dichotomy fails at all uncountable regular cardinals after we add a Cohen real to a model of GCH. We also discuss connections with some regularity properties, like the κ-perfect set property, the κ-Miller measurability, and the κ-Sacks measurability.

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Can we classify complete metric spaces up to isometry?

Ros

2017

Boll Unione Mat Ital

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We survey some old and new results concerning the classification of complete metric spaces up to isometry, a theme initiated by Gromov, Vershik and others. All theorems concerning separable spaces appeared in various papers in the last twenty years: here we tried to present them in a unitary and organic way, sometimes with new and/or simplified proofs. The results concerning non-separable spaces (and, to some extent, the setup and techniques used to handle them) are instead new, and suggest new lines of investigation in this area of research. Contents 1. Classification problems and Borel reducibility 2 2. The complexity of the isometry relation on separable spaces 10 3. Trees and isometry 26 4. The complexity of the isometry relation on non-separable spaces 34 5. Further extensions of the theory of Borel reducibility 41 References 42The purpose of this paper is threefold: (A) Give a concise and self-contained presentation of the theory of Borel reducibility for equivalence relations, including some motivations for its development, accessible to non-experts in the field (Section 1). (B) Illustrate how Borel reducibility (and descriptive set theory in general) may be used to gain new insight into some natural and interesting classification problems. We concentrate on the prominent example of classifying separable complete (i.e. Polish) metric spaces up to isometry, and survey the most important results in this area obtained by Gromov, Vershik, Kechris, Gao and many others, including some very recent progress on the long-standing open problem of determining the complexity of isometry between locally compact Polish metric spaces. When possible, we provide (sometimes simplified and self-contained) proofs of the most relevant results, or at least we overview the main ideas and methods involved in them (Sections 2 and 3).

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Continuous images of closed sets in generalized Baire spaces

Cited by 15 publications

References 17 publications

Questions on generalised Baire spaces

Questions on generalised Baire spaces

The Hurewicz dichotomy for generalized Baire spaces

Can we classify complete metric spaces up to isometry?

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