Analytic and coanalytic families of almost disjoint functions

2008 ◽  
Vol 73 (4) ◽  
pp. 1158-1172 ◽  
Author(s):  
Bart Kastermans ◽  
Juris Steprāns ◽  
Yi Zhang
Keyword(s):  
Analytic Family ◽  
Maximality Condition ◽  
Almost Disjoint

AbstractIf is an analytic family of pairwise eventually different functions then the following strong maximality condition fails: For any countable , no member of which is covered by finitely many functions from , there is such that for all there are infinitely many integers k such that f(k) = h(k). However if V = L then there exists a coanalytic family of pairwise eventually different functions satisfying this strong maximality condition.

scholarly journals Models of Set Theory in which Nonconstructible Reals First Appear at a Given Projective Level

Mathematics ◽  
2020 ◽  
Vol 8 (6) ◽  
pp. 910 ◽  
Author(s):  
Vladimir Kanovei ◽  
Vassily Lyubetsky
Keyword(s):  
Lower Limit ◽  
Set Theory ◽  
Generic Extension ◽  
Projective Class ◽  
Almost Disjoint ◽  
Projective Hierarchy ◽  
Models Of Set Theory

Models of set theory are defined, in which nonconstructible reals first appear on a given level of the projective hierarchy. Our main results are as follows. Suppose that n ≥ 2 . Then: 1. If it holds in the constructible universe L that a ⊆ ω and a ∉ Σ n 1 ∪ Π n 1 , then there is a generic extension of L in which a ∈ Δ n + 1 1 but still a ∉ Σ n 1 ∪ Π n 1 , and moreover, any set x ⊆ ω , x ∈ Σ n 1 , is constructible and Σ n 1 in L . 2. There exists a generic extension L in which it is true that there is a nonconstructible Δ n + 1 1 set a ⊆ ω , but all Σ n 1 sets x ⊆ ω are constructible and even Σ n 1 in L , and in addition, V = L [ a ] in the extension. 3. There exists an generic extension of L in which there is a nonconstructible Σ n + 1 1 set a ⊆ ω , but all Δ n + 1 1 sets x ⊆ ω are constructible and Δ n + 1 1 in L . Thus, nonconstructible reals (here subsets of ω ) can first appear at a given lightface projective class strictly higher than Σ 2 1 , in an appropriate generic extension of L . The lower limit Σ 2 1 is motivated by the Shoenfield absoluteness theorem, which implies that all Σ 2 1 sets a ⊆ ω are constructible. Our methods are based on almost-disjoint forcing. We add a sufficient number of generic reals to L , which are very similar at a given projective level n but discernible at the next level n + 1 .

Families of Almost Disjoint Sets

1974 ◽  
pp. 286-310
Author(s):  
W. Wistar Comfort ◽  
Stylianos Negrepontis
Keyword(s):  
Almost Disjoint ◽  
Disjoint Sets

ALMOST DISJOINT FAMILIES OF COUNTABLE SETS

1984 ◽  
pp. 59-88 ◽  
Author(s):  
B. BALCAR ◽  
J. DOČKÁLKOVÁ ◽  
P. SIMON
Keyword(s):  
Almost Disjoint ◽  
Almost Disjoint Families

scholarly journals Maximal metric surfaces and the Sobolev-to-Lipschitz property

2020 ◽  
Vol 59 (5) ◽  
Author(s):  
Paul Creutz ◽  
Elefterios Soultanis
Keyword(s):  
Metric Spaces ◽  
Equivalence Classes ◽  
Plateau Problem ◽  
Lipschitz Property ◽  
Maximality Condition ◽  
Volume Rigidity

Abstract We find maximal representatives within equivalence classes of metric spheres. For Ahlfors regular spheres these are uniquely characterized by satisfying the seemingly unrelated notions of Sobolev-to-Lipschitz property, or volume rigidity. We also apply our construction to solutions of the Plateau problem in metric spaces and obtain a variant of the associated intrinsic disc studied by Lytchak–Wenger, which satisfies a related maximality condition.

scholarly journals The Ramsey property implies no mad families

2019 ◽  
Vol 116 (38) ◽  
pp. 18883-18887 ◽  
Author(s):  
David Schrittesser ◽  
Asger Törnquist
Keyword(s):  
Set Theory ◽  
Large Set ◽  
Descriptive Set Theory ◽  
Ramsey Property ◽  
Mad Families ◽  
Almost Disjoint ◽  
Descriptive Set ◽  
Maximal Almost Disjoint ◽  
Borel Ideals ◽  
Standing Problem

We show that if all collections of infinite subsets of N have the Ramsey property, then there are no infinite maximal almost disjoint (mad) families. The implication is proved in Zermelo–Fraenkel set theory with only weak choice principles. This gives a positive solution to a long-standing problem that goes back to Mathias [A. R. D. Mathias, Ann. Math. Logic 12, 59–111 (1977)]. The proof exploits an idea which has its natural roots in ergodic theory, topological dynamics, and invariant descriptive set theory: We use that a certain function associated to a purported mad family is invariant under the equivalence relation E0 and thus is constant on a “large” set. Furthermore, we announce a number of additional results about mad families relative to more complicated Borel ideals.

scholarly journals ON THE STABILITY OF THE DIFFERENTIAL PROCESS GENERATED BY COMPLEX INTERPOLATION

2020 ◽  
pp. 1-32
Author(s):  
Jesús M. F. Castillo ◽  
Willian H. G. Corrêa ◽  
Valentin Ferenczi ◽  
Manuel González
Keyword(s):  
Unit Circle ◽  
Banach Spaces ◽  
Local Stability ◽  
Interpolation Method ◽  
Complex Interpolation ◽  
Analytic Family ◽  
Köthe Spaces ◽  
The Stability ◽  
Stability Results ◽  
Differential Process

We study the stability of the differential process of Rochberg and Weiss associated with an analytic family of Banach spaces obtained using the complex interpolation method for families. In the context of Köthe function spaces, we complete earlier results of Kalton (who showed that there is global bounded stability for pairs of Köthe spaces) by showing that there is global (bounded) stability for families of up to three Köthe spaces distributed in arcs on the unit circle while there is no (bounded) stability for families of four or more Köthe spaces. In the context of arbitrary pairs of Banach spaces, we present some local stability results and some global isometric stability results.

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