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Displaying similar documents to “Maximal almost disjoint families of functions”

Template iterations and maximal cofinitary groups

Vera Fischer, Asger Törnquist (2015)

Fundamenta Mathematicae

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Jörg Brendle (2003) used Hechler’s forcing notion for adding a maximal almost disjoint family along an appropriate template forcing construction to show that (the minimal size of a maximal almost disjoint family) can be of countable cofinality. The main result of the present paper is that g , the minimal size of a maximal cofinitary group, can be of countable cofinality. To prove this we define a natural poset for adding a maximal cofinitary group of a given cardinality, which enjoys...

Comparing the closed almost disjointness and dominating numbers

Dilip Raghavan, Saharon Shelah (2012)

Fundamenta Mathematicae

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We prove that if there is a dominating family of size ℵ₁, then there are ℵ₁ many compact subsets of ω ω whose union is a maximal almost disjoint family of functions that is also maximal with respect to infinite partial functions.

A MAD Q-set

Arnold W. Miller (2003)

Fundamenta Mathematicae

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A MAD (maximal almost disjoint) family is an infinite subset of the infinite subsets of ω = 0,1,2,... such that any two elements of intersect in a finite set and every infinite subset of ω meets some element of in an infinite set. A Q-set is an uncountable set of reals such that every subset is a relative G δ -set. It is shown that it is relatively consistent with ZFC that there exists a MAD family which is also a Q-set in the topology it inherits as a subset of P ( ω ) = 2 ω .

Supercompactness and partial level by level equivalence between strong compactness and strongness

Arthur W. Apter (2004)

Fundamenta Mathematicae

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We force and construct a model containing supercompact cardinals in which, for any measurable cardinal δ and any ordinal α below the least beth fixed point above δ, if δ + α is regular, δ is δ + α strongly compact iff δ is δ + α + 1 strong, except possibly if δ is a limit of cardinals γ which are δ + α strongly compact. The choice of the least beth fixed point above δ as our bound on α is arbitrary, and other bounds are possible.

Ordinal remainders of classical ψ-spaces

Alan Dow, Jerry E. Vaughan (2012)

Fundamenta Mathematicae

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Let ω denote the set of natural numbers. We prove: for every mod-finite ascending chain T α : α < λ of infinite subsets of ω, there exists [ ω ] ω , an infinite maximal almost disjoint family (MADF) of infinite subsets of the natural numbers, such that the Stone-Čech remainder βψ∖ψ of the associated ψ-space, ψ = ψ(ω,ℳ ), is homeomorphic to λ + 1 with the order topology. We also prove that for every λ < ⁺, where is the tower number, there exists a mod-finite ascending chain T α : α < λ , hence a ψ-space with...

MAD families and P -points

Salvador García-Ferreira, Paul J. Szeptycki (2007)

Commentationes Mathematicae Universitatis Carolinae

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The Katětov ordering of two maximal almost disjoint (MAD) families 𝒜 and is defined as follows: We say that 𝒜 K if there is a function f : ω ω such that f - 1 ( A ) ( ) for every A ( 𝒜 ) . In [Garcia-Ferreira S., Hrušák M., Ordering MAD families a la Katětov, J. Symbolic Logic 68 (2003), 1337–1353] a MAD family is called K -uniform if for every X ( 𝒜 ) + , we have that 𝒜 | X K 𝒜 . We prove that CH implies that for every K -uniform MAD family 𝒜 there is a P -point p of ω * such that the set of all Rudin-Keisler predecessors of p is dense...

Asymmetric tie-points and almost clopen subsets of *

Alan S. Dow, Saharon Shelah (2018)

Commentationes Mathematicae Universitatis Carolinae

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A tie-point of compact space is analogous to a cut-point: the complement of the point falls apart into two relatively clopen non-compact subsets. We review some of the many consistency results that have depended on the construction of tie-points of * . One especially important application, due to Veličković, was to the existence of nontrivial involutions on * . A tie-point of * has been called symmetric if it is the unique fixed point of an involution. We define the notion of an almost...

Characterization of Strongly Exposed Points in General Köthe-Bochner Banach Spaces

Houcine Benabdellah, My Hachem Lalaoui Rhali (2004)

Bulletin of the Polish Academy of Sciences. Mathematics

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We study strongly exposed points in general Köthe-Bochner Banach spaces X(E). We first give a characterization of strongly exposed points of the set of X-selections of a measurable multifunction Γ. We then apply this result to the study of strongly exposed points of the closed unit ball of X(E). Precisely we show that if an element f is a strongly exposed point of B X ( E ) , then |f| is a strongly exposed point of B X and f(ω)/∥ f(ω)∥ is a strongly exposed point of B E for μ-almost all ω ∈ S(f). ...

Another ⋄-like principle

Michael Hrušák (2001)

Fundamenta Mathematicae

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A new ⋄-like principle consistent with the negation of the Continuum Hypothesis is introduced and studied. It is shown that ¬ is consistent with CH and that in many models of = ω₁ the principle holds. As implies that there is a MAD family of size ℵ₁ this provides a partial answer to a question of J. Roitman who asked whether = ω₁ implies = ω₁. It is proved that holds in any model obtained by adding a single Laver real, answering a question of J. Brendle who asked whether = ω₁...

On families of Lindelöf and related subspaces of 2 ω

Lúcia Junqueira, Piotr Koszmider (2001)

Fundamenta Mathematicae

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We consider the families of all subspaces of size ω₁ of 2 ω (or of a compact zero-dimensional space X of weight ω₁ in general) which are normal, have the Lindelöf property or are closed under limits of convergent ω₁-sequences. Various relations among these families modulo the club filter in [ X ] ω are shown to be consistently possible. One of the main tools is dealing with a subspace of the form X ∩ M for an elementary submodel M of size ω₁. Various results with this flavor are obtained. Another...

Special sets of reals and weak forms of normality on Isbell--Mrówka spaces

Vinicius de Oliveira Rodrigues, Victor dos Santos Ronchim, Paul J. Szeptycki (2023)

Commentationes Mathematicae Universitatis Carolinae

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We recall some classical results relating normality and some natural weakenings of normality in Ψ -spaces over almost disjoint families of branches in the Cantor tree to special sets of reals like Q -sets, λ -sets and σ -sets. We introduce a new class of special sets of reals which corresponds to the corresponding almost disjoint family of branches being 0 -separated. This new class fits between λ -sets and perfectly meager sets. We also discuss conditions for an almost disjoint family 𝒜 being...

L-like Combinatorial Principles and Level by Level Equivalence

Arthur W. Apter (2009)

Bulletin of the Polish Academy of Sciences. Mathematics

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We force and construct a model in which GCH and level by level equivalence between strong compactness and supercompactness hold, along with certain additional “L-like” combinatorial principles. In particular, this model satisfies the following properties: (1) δ holds for every successor and Mahlo cardinal δ. (2) There is a stationary subset S of the least supercompact cardinal κ₀ such that for every δ ∈ S, δ holds and δ carries a gap 1 morass. (3) A weak version of δ holds for every...

On ω 2 -saturated families

Lajos Soukup (1991)

Commentationes Mathematicae Universitatis Carolinae

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If there is no inner model with measurable cardinals, then for each cardinal λ there is an almost disjoint family 𝒜 λ of countable subsets of λ such that every subset of λ with order type ω 2 contains an element of 𝒜 λ .

On some representations of almost everywhere continuous functions on m

Ewa Strońska (2006)

Colloquium Mathematicae

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It is proved that the following conditions are equivalent: (a) f is an almost everywhere continuous function on m ; (b) f = g + h, where g,h are strongly quasicontinuous on m ; (c) f = c + gh, where c ∈ ℝ and g,h are strongly quasicontinuous on m .