Complexity of Sets and Binary Relations in Continuum Theory: A Survey

Marcone, Alberto

doi:10.1007/3-7643-7692-9_5

Cited by 4 publications

(3 citation statements)

References 37 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…F is always Borel [6, Theorem 2], but besides that it can be arbitrarily complex [4, Fact 3.12]. Section 3.5 of [4] also gives us some examples:…”

Section: Saturated and Type-saturated Classesmentioning

confidence: 99%

“…Analogously to the saturated class we say that F ⊆ K([0, 1] ω ) is a saturated family if F = F Usually, when considering the complexity of a class C ⊆ K, the class is identified with max([C]) and its complexity in K([0, 1] ω ) is considered. There are many results on complexity of max([C]), see for example the survey [4]. We are rather interested in the lowest complexity among families in [C].…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Borel complexity up to the equivalence

Bartoš

2020

Topology and its Applications

View full text Add to dashboard Cite

We say that two classes of topological spaces are equivalent if each member of one class has a homeomorphic copy in the other class and vice versa. Usually when the Borel complexity of a class of metrizable compacta is considered, the class is realized as the subset of the hyperspace K([0, 1] ω ) containing all homeomorphic copies of members of the given class. We are rather interested in the lowest possible complexity among all equivalent realizations of the given class in the hyperspace.We recall that to every analytic subset of K([0, 1] ω ) there exist an equivalent G δ subset. Then we show that up to the equivalence open subsets of the hyperspace K([0, 1] ω ) correspond to countably many classes of metrizable compacta. Finally we use the structure of open subsets up to equivalence to prove that to every F σ subset of K([0, 1] ω ) there exists an equivalent closed subset.Classification: 54H05, 54B20, 54E45, 54F15.

show abstract

“…F is always Borel [6, Theorem 2], but besides that it can be arbitrarily complex [4, Fact 3.12]. Section 3.5 of [4] also gives us some examples:…”

Section: Saturated and Type-saturated Classesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Borel complexity up to the equivalence

Bartoš

2020

Topology and its Applications

View full text Add to dashboard Cite

show abstract

“…Example 4.31. By [9] the following classes are also coanalytically complete, and hence not strongly Polishable: hereditarily decomposable continua, dendroids, λ-dendroids, arcwise connected continua, uniquely arcwise connected continua, hereditarily locally connected continua.…”

Section: Proposition 414mentioning

confidence: 99%

Compactifiable classes of compacta

Bartoš

Bobok

Mill

et al. 2019

Topology and its Applications

View full text Add to dashboard Cite

We introduce the notion of compactifiable classes -these are classes of metrizable compact spaces that can be up to homeomorphic copies "disjointly combined" into one metrizable compact space. This is witnessed by so-called compact composition of the class. Analogously, we consider Polishable classes and Polish compositions. The question of compactifiability or Polishability of a class is related to hyperspaces. Strongly compactifiable and strongly Polishable classes may be characterized by the existence of a corresponding family in the hyperspace of all metrizable compacta. We systematically study the introduced notions -we give several characterizations, consider preservation under various constructions, and raise several questions.Classification: 54D80, 54H05, 54B20, 54E45, 54F15.

show abstract