Strong negative partition above the continuum

1990 ◽  
Vol 55 (1) ◽  
pp. 21-31 ◽  
Author(s):  
Saharon Shelah
Keyword(s):  
Boolean Algebras ◽  
Stationary Subset ◽  
Topological Spaces ◽  
Strong Limit ◽  
Equivalent Formulation ◽  
Detailed Proof ◽  
Preliminary Version ◽  
The Continuum

For e.g. λ = μ+, μ regular, λ larger than the continuum, we prove a strong nonpartition result (stronger than λ → [λ; λ]2). As a consequence, the product of two topological spaces of cellularity <λ may have cellularity λ, or, in equivalent formulation, the product of two λ-c.c. Boolean algebras may lack the λ-c.c. Also λ-S-spaces and λ-L-spaces exist. In fact we deal not with successors of regular λ but with regular λ above the continuum which has a nonreflecting stationary subset of ordinals with uncountable cofinalities; sometimes we require λ to be not strong limit.The paper is self-contained. On the nonpartition results see the closely related papers of Todorčević [T1], Shelah [Sh276] and [Sh261], and Shelah and Steprans [ShSt1].On the cellularity of products see Todorčević [T2] and [T3], where such results were obtained for (e.g.) cf and ; the class of cardinals he gets is quite disjoint from ours. In [Sh282] such results were obtained for more successors of singulars (mainly λ+, λ > 2cf λ). Also, concerning S and L spaces, Todorčević gets existence.Todorčević's work on cardinals like relies on [Sh68] (see more in [ShA2, Chapter XIII]) (the scales appearing in the proof of ). The problem was stressed in a preliminary version of the surveys of Juhasz and Monk. We give a detailed proof for one strong nonpartition theorem (1.1) and then give various strengthenings. We then use 1.10 to get the consequences (in 1.11 and 1.12).

Maximality in effective topology

1983 ◽  
Vol 48 (1) ◽  
pp. 100-112 ◽  
Author(s):  
Iraj Kalantari ◽  
Anne Leggett
Keyword(s):  
Topological Space ◽  
Boolean Algebras ◽  
Vector Spaces ◽  
Topological Spaces ◽  
Mathematical Structures ◽  
Topological Vector Spaces ◽  
Linear Orderings ◽  
Recursively Enumerable ◽  
Countable Basis ◽  
Effective Analysis

In this paper we continue the study of the structure of the lattice of recursively enumerable (r.e.) open subsets of a topological space. Work in this approach to effective topology began in Kalantari and Retzlaff [5] and continued in Kalantari [2], Kalantari and Leggett [3] and Kalantari and Remmel [4]. Studies in effectiveness of results in structures other than integers began with the work of Specker [17] and Lacombe [8] on effective analysis.The renewed activity in the study of the effective content of mathematical structures owes much to Nerode's program and Metakides' and Nerode's [11], [12] work on vector spaces and fields. These studies have been extended by Kalantari, Remmel, Retzlaff, Shore and Smith. Similar studies on the effective content of other mathematical structures have been conducted. These include work on topological vector spaces, boolean algebras, linear orderings etc.Kalantari and Retzlaff [5] began a study of effective topological spaces by considering a topological space with a countable basis ⊿ for the topology. The space X is to be fully effective; that is, the basis elements are coded into ω and the operations of intersection of basis elements and the relation of inclusion among them are both computable. An r.e. open subset of X is then represented as the union of basic open sets whose codes lie in an r.e. subset of ω.

NOWHERE PRECIPITOUSNESS OF THE NON-STATIONARY IDEAL OVER ${\mathcal P}_\kappa \lambda$

2002 ◽  
Vol 02 (01) ◽  
pp. 81-89 ◽  
Author(s):  
YO MATSUBARA ◽  
SAHARON SHELAH
Keyword(s):  
Stationary Subset ◽  
Singular Cardinal ◽  
Strong Limit ◽  
Uncountable Cardinal

We prove that if λ is a strong limit singular cardinal and κ a regular uncountable cardinal < λ, then NSκλ, the non-stationary ideal over [Formula: see text], is nowhere precipitous. We also show that under the same hypothesis every stationary subset of [Formula: see text] can be partitioned into λκ disjoint stationary sets.

Continuum-many Boolean algebras of the form Borel

2004 ◽  
Vol 69 (3) ◽  
pp. 799-816 ◽  
Author(s):  
Michael Ray Oliver
Keyword(s):  
Equivalence Relation ◽  
Continuum Hypothesis ◽  
Boolean Algebras ◽  
New Technique ◽  
Finite Sets ◽  
A New Technique ◽  
Borel Ideals ◽  
Forcing Axiom ◽  
The Continuum ◽  
The Ideal

Abstract.We examine the question of how many Boolean algebras, distinct up to isomorphism, that are quotients of the powerset of the naturals by Borel ideals, can be proved to exist in ZFC alone. The maximum possible value is easily seen to be the cardinality of the continuum ; earlier work by Ilijas Farah had shown that this was the value in models of Martin's Maximum or some similar forcing axiom, but it was open whether there could be fewer in models of the Continuum Hypothesis.We develop and apply a new technique for constructing many ideals whose quotients must be nonisomorphic in any model of ZFC. The technique depends on isolating a kind of ideal, called shallow, that can be distinguished from the ideal of all finite sets even after any isomorphic embedding, and then piecing together various copies of the ideal of all finite sets using distinct shallow ideals. In this way we are able to demonstrate that there are continuum-many distinct quotients by Borel ideals, indeed by analytic P-ideals, and in fact that there is in an appropriate sense a Borel embedding of the Vitali equivalence relation into the equivalence relation of isomorphism of quotients by analytic P-ideals. We also show that there is an uncountable definable wellordered collection of Borel ideals with distinct quotients.

scholarly journals Noncommutative Lattices and the Algebras of Their Continuous Functions

1998 ◽  
Vol 10 (04) ◽  
pp. 439-466 ◽  
Author(s):  
Elisa Ercolessi ◽  
Giovanni Landi ◽  
Paulo Teotonio-Sobrinho
Keyword(s):  
Quantum Physics ◽  
Continuous Functions ◽  
Topological Spaces ◽  
Hausdorff Topology ◽  
Topological Information ◽  
C Algebras ◽  
Finite Dimensional ◽  
Noncommutative Algebras ◽  
Af Algebras ◽  
The Continuum

Recently a new kind of approximation to continuum topological spaces has been introduced, the approximating spaces being partially ordered sets (posets) with a finite or at most a countable number of points. The partial order endows a poset with a nontrivial non-Hausdorff topology. Their ability to reproduce important topological information of the continuum has been the main motivation for their use in quantum physics. Posets are truly noncommutative spaces, or noncommutative lattices, since they can be realized as structure spaces of noncommutative C*-algebras. These noncommutative algebras play the same rôle as the algebra of continuous functions [Formula: see text] on a Hausdorff topological space M and can be thought of as algebras of operator valued functions on posets. In this article, we will review some mathematical results that establish a duality between finite posets and a certain class of C*-algebras. We will see that the algebras in question are all postliminal approximately finite dimensional (AF) algebras.

On uniform boundedness properties in exhausting additive set function spaces

1981 ◽  
Vol 90 (1-2) ◽  
pp. 175-184 ◽  
Author(s):  
Aníbal Moltó
Keyword(s):  
Topological Group ◽  
Duality Theory ◽  
Uniform Boundedness ◽  
Baire Category ◽  
Boolean Algebras ◽  
Topological Spaces ◽  
Baire Category Theorem ◽  
Set Functions ◽  
Category Theorem ◽  
Set Function

SynopsisValdivia (1978) introduced the class of suprabarrelled spaces, and (1979) deduced some uniform boundedness properties for scalar valued exhausting additive set functions on a σ-algebra from the suprabarrelledness of certain spaces. In this paper, it is shown that those uniform boundedness properties hold for G-valued exhausting additive set functions, G being a commutative topological group, on a larger class of Boolean algebras. Such properties are proved in Valdivia (1979) by means of duality theory arguments and ‘sliding hump’ methods, whereas here they are derived from the Baire category theorem. This generalization enables us to find a wide class of compact topological spaces K such that the subspaces of C(K) which satisfy a mild property are suprabarrelled.

scholarly journals A game on Boolean algebras describing the collapse of the continuum

2009 ◽  
Vol 160 (1) ◽  
pp. 117-126
Author(s):  
Miloš S. Kurilić ◽  
Boris Šobot
Keyword(s):  
Boolean Algebras ◽  
The Continuum

scholarly journals σ-Continuous Functions and Related Cardinal Characteristics of the Continuum

2020 ◽  
Vol 76 (1) ◽  
pp. 1-10
Author(s):  
Taras Banakh
Keyword(s):  
Continuous Functions ◽  
Topological Spaces ◽  
Cardinal Characteristics ◽  
The Continuum

AbstractA function f : X → Y between topological spaces is called σ-continuous (resp. ̄σ-continuous) if there exists a (closed) cover {Xn}n∈ω of X such that for every n ∈ ω the restriction f ↾ Xn is continuous. By 𝔠 σ (resp. 𝔠¯σ)we denote the largest cardinal κ ≤ 𝔠 such that every function f : X → ℝ defined on a subset X ⊂ ℝ of cardinality |X| <κ is σ-continuous (resp. ¯σ-continuous). It is clear that ω1 ≤ 𝔠¯σ ≤ 𝔠 σ ≤ 𝔠.We prove that 𝔭 ≤ 𝔮0 = 𝔠¯σ =min{𝔠 σ, 𝔟, 𝔮 }≤ 𝔠 σ ≤ min{non(ℳ), non(𝒩)}.

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