scholarly journals Stationary subspaces in ordered spaces

1989 ◽  
Vol 40 (3) ◽  
pp. 381-387
Author(s):  
Nobuyuki Kemoto
Keyword(s):  
Regular Cardinal ◽  
Closed Subspace ◽  
Stationary Set

In this paper, we shall characterise the B(k)-property in generalised ordered (GO) spaces as follows.For every uncountable regular cardinal K, every GO space has the B(K)-property if and only if it has no closed subspace which is homeomorphic to a stationary set in K (with the subspace topology in K).

Increasing u2 by a stationary set preserving forcing

2009 ◽  
Vol 74 (1) ◽  
pp. 187-200
Author(s):  
Benjamin Claverie ◽  
Ralf Schindler
Keyword(s):  
Regular Cardinal ◽  
Nonstationary Ideal ◽  
Stationary Set ◽  
Precipitous Ideal ◽  
Countable Structure

AbstractWe show that if I is a precipitous ideal on ω1 and if θ > ω1 is a regular cardinal, then there is a forcing ℙ = ℙ(I, θ) which preserves the stationarity of all I-positive sets such that in Vℙ, ⟨Hθ; ∈, I⟩ is a generic iterate of a countable structure ⟨M; ∈, Ī⟩. This shows that if the nonstationary ideal on ω1 is precipitous and exists, then there is a stationary set preserving forcing which increases . Moreover, if Bounded Martin's Maximum holds and the nonstationary ideal on ω1 is precipitous, then .

Paracompactness for Ordered Sums

1978 ◽  
Vol 21 (3) ◽  
pp. 363-364
Author(s):  
Isidore Fleischer
Keyword(s):  
Regular Cardinal ◽  
Closed Subspace ◽  
Ordered Set ◽  
Regular Cardinals ◽  
Linearly Ordered Set

Let us say that an order embedding of an uncountable regular cardinal in a linearly ordered set is continuous if it preserves the suprema (for all smaller limit ordinals). This makes the embedding a homeomorphism for the two order topologies; and if the image has no supremum, it is a closed subspace. Since uncountable regular cardinals fail to be paracompact, a linearly ordered set can be paracompact only if it admits no such embedding or anti-embedding. Conversely, Gillman and Henriksen have shown that this suffices (Trans. A.M.S. 77 (1954) pp. 352 ff).

On a topological construction of Juhasz and Shelah

1992 ◽  
Vol 57 (1) ◽  
pp. 166-171
Author(s):  
Dan Velleman
Keyword(s):  
Topological Space ◽  
Additional Structure ◽  
Homogeneous Set ◽  
Stationary Set ◽  
Topological Construction

In [2], Juhasz and Shelah use a forcing argument to show that it is consistent with GCH that there is a 0-dimensional T2 topological space X of cardinality ℵ3 such that every partition of the triples of X into countably many pieces has a nondiscrete (in the topology) homogeneous set. In this paper we will show how to construct such a space using a simplified (ω2, 1)-morass with certain additional structure added to it. The additional structure will be a slight strengthening of a built-in ◊ sequence, analogous to the strengthening of ordinary ◊k to ◊S for a stationary set S ⊆ k.Suppose 〈〈θα∣ ∝ ≤ ω2〉, 〈∝β∣α < β ≤ ω2〉〉 is a neat simplified (ω2, 1)-morass (see [3]). Let ℒ be a language with countably many symbols of all types, and suppose that for each α < ω2, α is an ℒ-structure with universe θα. The sequence 〈α∣α < ω2 is called a built-in ◊ sequence for the morass if for every ℒ-structure with universe ω3 there is some α < ω2 and some f ∈αω2 such that f(α) ≺ , where f(α) is the ℒ-structure isomorphic to α under the isomorphism f. We can strengthen this slightly by assuming that α is only defined for α ∈ S, for some stationary set S ⊆ ω2. We will then say that is a built-in ◊ sequence on levels in S if for every ℒ-structure with universe ω3 there is some α ∈ S and some f ∈ αω2 such that f(α) ≺ .

scholarly journals In what spaces is every closed normal cone regular?

1970 ◽  
Vol 17 (2) ◽  
pp. 121-125 ◽  
Author(s):  
C. W. McArthur
Keyword(s):  
Banach Space ◽  
Convex Space ◽  
Unconditional Basis ◽  
Normal Cone ◽  
Closed Subspace ◽  
Locally Convex Space ◽  
Regular Part ◽  
Fréchet Space ◽  
Sequential Completeness ◽  
Locally Convex

It is known (13, p. 92) that each closed normal cone in a weakly sequentially complete locally convex space is regular and fully regular. Part of the main theorem of this paper shows that a certain amount of weak sequential completeness is necessary in order that each closed normal cone be regular. Specifically, it is shown that each closed normal cone in a Fréchet space is regular if and only if each closed subspace with an unconditional basis is weakly sequentially complete. If E is a strongly separable conjugate of a Banach space it is shown that each closed normal cone in E is fully regular. If E is a Banach space with an unconditional basis it is shown that each closed normal cone in E is fully regular if and only if E is the conjugate of a Banach space.

Generic Σ31 absoluteness

2004 ◽  
Vol 69 (1) ◽  
pp. 73-80 ◽  
Author(s):  
Sy D. Friedman
Keyword(s):  
Regular Cardinal ◽  
Forcing Axioms ◽  
Proper Forcing ◽  
Class Forcing ◽  
The Universe ◽  
Generic Extensions

In this article we study the strength of absoluteness (with real parameters) in various types of generic extensions, correcting and improving some results from [3]. (In particular, see Theorem 3 below.) We shall also make some comments relating this work to the bounded forcing axioms BMM, BPFA and BSPFA.The statement “ absoluteness holds for ccc forcing” means that if a formula with real parameters has a solution in a ccc set-forcing extension of the universe V, then it already has a solution in V. The analogous definition applies when ccc is replaced by other set-forcing notions, or by class-forcing.Theorem 1. [1] absoluteness for ccc has no strength; i.e., if ZFC is consistent then so is ZFC + absoluteness for ccc.The following results concerning (arbitrary) set-forcing and class-forcing can be found in [3].Theorem 2 (Feng-Magidor-Woodin). (a) absoluteness for arbitrary set-forcing is equiconsistent with the existence of a reflecting cardinal, i.e., a regular cardinal κ such that H(κ) is ∑2-elementary in V.(b) absoluteness for class-forcing is inconsistent.We consider next the following set-forcing notions, which lie strictly between ccc and arbitrary set-forcing: proper, semiproper, stationary-preserving and ω1-preserving. We refer the reader to [8] for the definitions of these forcing notions.Using a variant of an argument due to Goldstern-Shelah (see [6]), we show the following. This result corrects Theorem 2 of [3] (whose proof only shows that if absoluteness holds in a certain proper forcing extension, then in L either ω1 is Mahlo or ω2 is inaccessible).

scholarly journals On Spectral Properties of Compact Toeplitz Operators on Bergman Space with Logarithmically Decaying Symbol and Applications to Banded Matrices

2021 ◽  
Author(s):  
Mahamet Koïta ◽  
Stanislas Kupin ◽  
Sergey Naboko ◽  
Belco Touré
Keyword(s):  
Spectral Analysis ◽  
Continuous Function ◽  
Bergman Space ◽  
Orthogonal Projection ◽  
Spectral Properties ◽  
Toeplitz Operators ◽  
Closed Subspace ◽  
Jacobi Matrices ◽  
Banded Matrices ◽  
Eigen Values

Abstract Let $L^2({{\mathbb{D}}})$ be the space of measurable square-summable functions on the unit disk. Let $L^2_a({{\mathbb{D}}})$ be the Bergman space, that is, the (closed) subspace of analytic functions in $L^2({{\mathbb{D}}})$. $P_+$ stays for the orthogonal projection going from $L^2({{\mathbb{D}}})$ to $L^2_a({{\mathbb{D}}})$. For a function $\varphi \in L^\infty ({{\mathbb{D}}})$, the Toeplitz operator $T_\varphi : L^2_a({{\mathbb{D}}})\to L^2_a({{\mathbb{D}}})$ is defined as $$\begin{align*} & T_\varphi f=P_+\varphi f, \quad f\in L^2_a({{\mathbb{D}}}). \end{align*}$$The main result of this article are spectral asymptotics for singular (or eigen-) values of compact Toeplitz operators with logarithmically decaying symbols, that is, $$\begin{align*} & \varphi(z)=\varphi_1(e^{i\theta})\, (1+\log(1/(1-r)))^{-\gamma},\quad \gamma&gt;0, \end{align*}$$where $z=re^{i\theta }$ and $\varphi _1$ is a continuous (or piece-wise continuous) function on the unit circle. The result is applied to the spectral analysis of banded (including Jacobi) matrices.

Constructible lattices of c-degrees

1982 ◽  
Vol 47 (4) ◽  
pp. 739-754
Author(s):  
C.P. Farrington
Keyword(s):  
Set Theory ◽  
Regular Cardinal ◽  
Boolean Algebras ◽  
Combinatorial Complexity ◽  
Generic Extension ◽  
Transitive Model ◽  
Perfect Set ◽  
Consistency Results ◽  
Combinatorial Principles ◽  
Souslin Trees

This paper is devoted to the proof of the following theorem.Theorem. Let M be a countable standard transitive model of ZF + V = L, and let ℒ Є M be a wellfounded lattice in M, with top and bottom. Let ∣ℒ∣M = λ, and suppose κ ≥ λ is a regular cardinal in M. Then there is a generic extension N of M such that(i) N and M have the same cardinals, and κN ⊂ M;(ii) the c-degrees of sets of ordinals of N form a pattern isomorphic to ℒ;(iii) if A ⊂ On and A Є N, there is B Є P(κ+)N such that L(A) = L(B).The proof proceeds by forcing with Souslin trees, and relies heavily on techniques developed by Jech. In [5] he uses these techniques to construct simple Boolean algebras in L, and in [6] he uses them to construct a model of set theory whose c-degrees have orderlype 1 + ω*.The proof also draws on ideas of Adamovicz. In [1]–[3] she obtains consistency results concerning the possible patterns of c-degrees of sets of ordinals using perfect set forcing and symmetric models. These methods have the advantage of yielding real degrees, but involve greater combinatorial complexity, in particular the use of ‘sequential representations’ of lattices.The advantage of the approach using Souslin trees is twofold: first, we can make use of ready-made combinatorial principles which hold in L, and secondly, the notion of genericity over a Souslin tree is particularly simple.

Powers of the ideal of Lebesgue measure zero sets

1991 ◽  
Vol 56 (1) ◽  
pp. 103-107
Author(s):  
Maxim R. Burke
Keyword(s):  
Partial Order ◽  
Lebesgue Measure ◽  
Regular Cardinal ◽  
Measure Zero ◽  
Lebesgue Measure Zero ◽  
Zero Sets ◽  
Covering Lemma ◽  
Inner Model ◽  
The Ideal

AbstractWe investigate the cofinality of the partial order κ of functions from a regular cardinal κ into the ideal of Lebesgue measure zero subsets of R. We show that when add () = κ and the covering lemma holds with respect to an inner model of GCH, then cf (κ) = max{cf(κκ), cf([cf()]κ)}. We also give an example to show that the covering assumption cannot be removed.

scholarly journals Best Simultaneous Approximation in Orlicz Spaces

2007 ◽  
Vol 2007 ◽  
pp. 1-7 ◽  
Author(s):  
M. Khandaqji ◽  
Sh. Al-Sharif
Keyword(s):  
Banach Space ◽  
Simultaneous Approximation ◽  
Closed Subspace ◽  
Orlicz Spaces ◽  
Unit Interval ◽  
Luxemburg Norm ◽  
Integrable Functions ◽  
Best Simultaneous Approximation ◽  
Distance Formula

LetXbe a Banach space and letLΦ(I,X)denote the space of OrliczX-valued integrable functions on the unit intervalIequipped with the Luxemburg norm. In this paper, we present a distance formula dist(f1,f2,LΦ(I,G))Φ, whereGis a closed subspace ofX, andf1,f2∈LΦ(I,X). Moreover, some related results concerning best simultaneous approximation inLΦ(I,X)are presented.

scholarly journals Characterizing Complete Erdős Space

2009 ◽  
Vol 61 (1) ◽  
pp. 124-140 ◽  
Author(s):  
Jan J. Dijkstra ◽  
Jan van Mill
Keyword(s):  
Hilbert Space ◽  
Banach Spaces ◽  
Closed Subspace ◽  
Convergent Sequence ◽  
Cantor Set

Abstract. The space now known as complete Erdős space was introduced by Paul Erdős in 1940 as the closed subspace of the Hilbert space ℓ2 consisting of all vectors such that every coordinate is in the convergent sequence ﹛0﹜ ∪ ﹛1/n : n ∈ℕ﹜. In a solution to a problem posed by Lex G. Oversteegen we present simple and useful topological characterizations of . As an application we determine the class of factors of . In another application we determine precisely which of the spaces that can be constructed in the Banach spaces ℓp according to the ‘Erdős method’ are homeomorphic to . A novel application states that if I is a Polishable Fσ-ideal on ω, then I with the Polish topology is homeomorphic to either ℤ, the Cantor set 2ω, ℤ × 2ω, or . This last result answers a question that was asked by Stevo Todorčević.

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