Two remarks about analytic sets

Engelen, Fons van; Kunen, Kenneth; Miller, Arnold W.

doi:10.1007/bfb0097332

Cited by 9 publications

(8 citation statements)

References 2 publications

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“…Using a similar argument and the Silver Theorem [10, 35.20] one can show that ideals generated by an coanalytic equivalence relation with uncountable many equivalence classes (i.e., ideals of sets which can be covered by countably many equivalence classes) have property (M). By 14 the same holds for the ideal of subsets of the plane which can be covered by countably many lines. There are other σ‐ideals with property (M): ideals on Polish spaces with a \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\Sigma ^0_2$\end{document} base which are not ccc (see 12), some ideals defined by translations (see 2 and 13).…”

Section: Fubini Productsmentioning

confidence: 77%

Ideals with bases of unbounded Borel complexity

Borodulin–Nadzieja

Głąb

2011

Mathematical Logic Quarterly

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Section: Fubini Productsmentioning

confidence: 77%

Ideals with bases of unbounded Borel complexity

Borodulin–Nadzieja

Głąb

2011

Mathematical Logic Quarterly

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“…In [3] it was proved that if an analytic set on the real plane cannot be covered by countably many lines then it contains a perfect set which also cannot be covered by countably many lines. We can generalize this in the following.…”

Section: Examplementioning

confidence: 98%

“…X / ∈ J -in this case we say that (n, F )-system S is proper. 3 be a set of all non-collinear triples. Then Y is a Polish subspace as an open subset of (X) 3 :…”

mentioning

confidence: 99%

LK property for σ-ideals

Głąb

2011

Topology and its Applications

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“…All these cases define clopen neighbourhoods of 1 A disjoint from {1 B : B ∈ A}, as required. Now we step up the cardinality dichotomy for Borel subsets of 2 N to its square 2 N × 2 N , using a result from [18] of van Engelen, Kunen and Miller. Lemma 31.…”

Section: Borel Structure Matrices and Almost Disjoint Familiesmentioning

confidence: 99%

A Banach space induced by an almost disjoint family, admitting only few operators and decompositions

Koszmider

Laustsen

2021

Advances in Mathematics

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We consider the closed subspace of ∞ generated by c 0 and the characteristic functions of elements of an uncountable, almost disjoint family A of infinite subsets of N. This Banach space has the form C 0 (K A ) for a locally compact Hausdorff space K A that is known under many names, including Ψ-space and Isbell-Mrówka space.We construct an uncountable, almost disjoint family A such that the algebra of all bounded linear operators on C 0 (K A ) is as small as possible in the precise sense that every bounded linear operator on C 0 (K A ) is the sum of a scalar multiple of the identity and an operator that factors through c 0 (which in this case is equivalent to having separable range). This implies that C 0 (K A ) has the fewest possible decompositions: whenever C 0 (K A ) is written as the direct sum of two infinite-dimensional Banach spaces X and Y, either X is isomorphic to C 0 (K A ) and Y to c 0 , or vice versa. These results improve previous work of the first named author in which an extra set-theoretic hypothesis was required. We also discuss the consequences of these results for the algebra of all bounded linear operators on our Banach space C 0 (K A ) concerning the lattice of closed ideals, characters and automatic continuity of homomorphisms.To exploit the perfect set property for Borel sets as in the classical construction of an almost disjoint family by Mrówka, we need to deal with N × N matrices rather than with the usual partitioners of an almost disjoint family. This noncommutative setting requires new ideas inspired by the theory of compact and weakly compact operators and the use of an extraction principle due to van Engelen, Kunen and Miller concerning Borel subsets of the square.

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Two remarks about analytic sets

Cited by 9 publications

References 2 publications

Ideals with bases of unbounded Borel complexity

Ideals with bases of unbounded Borel complexity

LK property for σ-ideals

A Banach space induced by an almost disjoint family, admitting only few operators and decompositions

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