On the complexity of subspaces of $S_{ω}$

Carlos Uzcátegui

Displaying similar documents to “On the complexity of subspaces of $S_{ω}$ ”

Infinite-Dimensionality modulo Absolute Borel Classes

Vitalij Chatyrko, Yasunao Hattori (2008)

Bulletin of the Polish Academy of Sciences. Mathematics

Similarity:

For each ordinal 1 ≤ α < ω₁ we present separable metrizable spaces $X_{α}$ , $Y_{α}$ and $Z_{α}$ such that (i) $f X_{α}, f Y_{α}, f Z_{α} = ω ₀$ , where f is either trdef or ₀-trsur, (ii) $A (α) - t r i n d X_{α} = \infty$ and $M (α) - t r i n d X_{α} = - 1$ , (iii) $A (α) - t r i n d Y_{α} = - 1$ and $M (α) - t r i n d Y_{α} = \infty$ , and (iv) $A (α) - t r i n d Z_{α} = M (α) - t r i n d Z_{α} = \infty$ and $A (α + 1) \cap M (α + 1) - t r i n d Z_{α} = - 1$ . We also show that there exists no separable metrizable space $W_{α}$ with $A (α) - t r i n d W_{α} \neq \infty$ , $M (α) - t r i n d W_{α} \neq \infty$ and $A (α) \cap M (α) - t r i n d W_{α} = \infty$ , where A(α) (resp. M(α)) is the absolutely additive (resp. multiplicative) Borel class.

Decomposing Borel functions using the Shore-Slaman join theorem

Takayuki Kihara (2015)

Fundamenta Mathematicae

Similarity:

Jayne and Rogers proved that every function from an analytic space into a separable metrizable space is decomposable into countably many continuous functions with closed domains if and only if the preimage of each $F_{σ}$ set under that function is again $F_{σ}$ . Many researchers conjectured that the Jayne-Rogers theorem can be generalized to all finite levels of Borel functions. In this paper, by using the Shore-Slaman join theorem on the Turing degrees, we show the following variant of the Jayne-Rogers...

Failure of the Factor Theorem for Borel pre-Hilbert spaces

Tadeusz Dobrowolski, Witold Marciszewski (2002)

Fundamenta Mathematicae

Similarity:

In every infinite-dimensional Fréchet space X, we construct a linear subspace E such that E is an $F_{σ δ σ}$ -subset of X and contains a retract R so that $R \times E^{ω}$ is not homeomorphic to $E^{ω}$ . This shows that Toruńczyk’s Factor Theorem fails in the Borel case.

Hyperspaces of Finite Sets in Universal Spaces for Absolute Borel Classes

Kotaro Mine, Katsuro Sakai, Masato Yaguchi (2005)

Bulletin of the Polish Academy of Sciences. Mathematics

Similarity:

By Fin(X) (resp. $F i n^{k} (X)$ ), we denote the hyperspace of all non-empty finite subsets of X (resp. consisting of at most k points) with the Vietoris topology. Let ℓ₂(τ) be the Hilbert space with weight τ and $ℓ ₂^{f} (τ)$ the linear span of the canonical orthonormal basis of ℓ₂(τ). It is shown that if $E = ℓ ₂^{f} (τ)$ or E is an absorbing set in ℓ₂(τ) for one of the absolute Borel classes $_{α} (τ)$ and $_{α} (τ)$ of weight ≤ τ (α > 0) then Fin(E) and each $F i n^{k} (E)$ are homeomorphic to E. More generally, if X is a connected E-manifold then Fin(X)...

On a problem concerning quasianalytic local rings

Hassan Sfouli (2014)

Annales Polonici Mathematici

Similarity:

Let (ₙ)ₙ be a quasianalytic differentiable system. Let m ∈ ℕ. We consider the following problem: let $f \in_{m}$ and f̂ be its Taylor series at $0 \in ℝ^{m}$ . Split the set $ℕ^{m}$ of exponents into two disjoint subsets A and B, $ℕ^{m} = A \cup B$ , and decompose the formal series f̂ into the sum of two formal series G and H, supported by A and B, respectively. Do there exist $g, h \in_{m}$ with Taylor series at zero G and H, respectively? The main result of this paper is the following: if we have a positive answer to the above problem for some...

Borel classes of uniformizations of sets with large sections

Petr Holický (2010)

Fundamenta Mathematicae

Similarity:

We give several refinements of known theorems on Borel uniformizations of sets with “large sections”. In particular, we show that a set B ⊂ [0,1] × [0,1] which belongs to $Σ ⁰_{α}$ , α ≥ 2, and which has all “vertical” sections of positive Lebesgue measure, has a $Π ⁰_{α}$ uniformization which is the graph of a $Σ ⁰_{α}$ -measurable mapping. We get a similar result for sets with nonmeager sections. As a corollary we derive an improvement of Srivastava’s theorem on uniformizations for Borel sets with $G_{δ}$ sections. ...

On Hattori spaces

A. Bouziad, E. Sukhacheva (2017)

Commentationes Mathematicae Universitatis Carolinae

Similarity:

For a subset $A$ of the real line $ℝ$ , Hattori space $H (A)$ is a topological space whose underlying point set is the reals $ℝ$ and whose topology is defined as follows: points from $A$ are given the usual Euclidean neighborhoods while remaining points are given the neighborhoods of the Sorgenfrey line. In this paper, among other things, we give conditions on $A$ which are sufficient and necessary for $H (A)$ to be respectively almost Čech-complete, Čech-complete, quasicomplete, Čech-analytic and weakly separated...

An irrational problem

Franklin D. Tall (2002)

Fundamenta Mathematicae

Similarity:

Given a topological space ⟨X,⟩ ∈ M, an elementary submodel of set theory, we define $X_{M}$ to be X ∩ M with topology generated by $U \cap M : U \in \cap M$ . Suppose $X_{M}$ is homeomorphic to the irrationals; must $X = X_{M}$ ? We have partial results. We also answer a question of Gruenhage by showing that if $X_{M}$ is homeomorphic to the “Long Cantor Set”, then $X = X_{M}$ .

Complete pairs of coanalytic sets

Jean Saint Raymond (2007)

Fundamenta Mathematicae

Similarity:

Let X be a Polish space, and let C₀ and C₁ be disjoint coanalytic subsets of X. The pair (C₀,C₁) is said to be complete if for every pair (D₀,D₁) of disjoint coanalytic subsets of $ω^{ω}$ there exists a continuous function $f : ω^{ω} \to X$ such that $f^{- 1} (C ₀) = D ₀$ and $f^{- 1} (C ₁) = D ₁$ . We give several explicit examples of complete pairs of coanalytic sets.

More reflections on compactness

Lúcia R. Junqueira, Franklin D. Tall (2003)

Fundamenta Mathematicae

Similarity:

We consider the question of when $X_{M} = X$ , where $X_{M}$ is the elementary submodel topology on X ∩ M, especially in the case when $X_{M}$ is compact.

A pure smoothness condition for Radó’s theorem for $α$ -analytic functions

Abtin Daghighi, Frank Wikström (2016)

Czechoslovak Mathematical Journal

Similarity:

Let $Ω \subset ℂ^{n}$ be a bounded, simply connected $ℂ$ -convex domain. Let $α \in ℤ_{+}^{n}$ and let $f$ be a function on $Ω$ which is separately $C^{2 α_{j} - 1}$ -smooth with respect to $z_{j}$ (by which we mean jointly $C^{2 α_{j} - 1}$ -smooth with respect to $Re z_{j}$ , $Im z_{j}$ ). If $f$ is $α$ -analytic on $Ω ∖ f^{- 1} (0)$ , then $f$ is $α$ -analytic on $Ω$ . The result is well-known for the case $α_{i} = 1$ , $1 \leq i \leq n$ , even when $f$ a priori is only known to be continuous.

$Σ_{s}$ -products revisited

Reynaldo Rojas-Hernández (2015)

Commentationes Mathematicae Universitatis Carolinae

Similarity:

We show that any $Σ_{s}$ -product of at most $𝔠$ -many $L Σ (\leq ω)$ -spaces has the $L Σ (\leq ω)$ -property. This result generalizes some known results about $L Σ (\leq ω)$ -spaces. On the other hand, we prove that every $Σ_{s}$ -product of monotonically monolithic spaces is monotonically monolithic, and in a similar form, we show that every $Σ_{s}$ -product of Collins-Roscoe spaces has the Collins-Roscoe property. These results generalize some known results about the Collins-Roscoe spaces and answer some questions due to Tkachuk [Lifting the Collins-Roscoe...

The "Full Clarkson-Erdős-Schwartz Theorem" on the closure of non-dense Müntz spaces

Tamás Erdélyi (2003)

Studia Mathematica

Similarity:

Denote by spanf₁,f₂,... the collection of all finite linear combinations of the functions f₁,f₂,... over ℝ. The principal result of the paper is the following. Theorem (Full Clarkson-Erdős-Schwartz Theorem). Suppose ${(λ_{j})}_{j = 1}^{\infty}$ is a sequence of distinct positive numbers. Then $s p a n 1, x^{λ ₁}, x^{λ ₂}, . . .$ is dense in C[0,1] if and only if $\sum_{j = 1}^{\infty} (λ_{j}) / (λ_{j} ² + 1) = \infty$ . Moreover, if $\sum_{j = 1}^{\infty} (λ_{j}) / (λ_{j} ² + 1) < \infty$ , then every function from the C[0,1] closure of $s p a n 1, x^{λ ₁}, x^{λ ₂}, . . .$ can be represented as an analytic function on z ∈ ℂ ∖ (-∞, 0]: |z| < 1 restricted to (0,1). This result improves an...

Cardinalities of DCCC normal spaces with a rank 2-diagonal

Wei-Feng Xuan, Wei-Xue Shi (2016)

Mathematica Bohemica

Similarity:

A topological space $X$ has a rank 2-diagonal if there exists a diagonal sequence on $X$ of rank $2$ , that is, there is a countable family ${𝒰_{n} : n \in ω}$ of open covers of $X$ such that for each $x \in X$ , ${x} = ⋂ {{St}^{2} (x, 𝒰_{n}) : n \in ω}$ . We say that a space $X$ satisfies the Discrete Countable Chain Condition (DCCC for short) if every discrete family of nonempty open subsets of $X$ is countable. We mainly prove that if $X$ is a DCCC normal space with a rank 2-diagonal, then the cardinality of $X$ is at most $𝔠$ . Moreover, we prove that if $X$ is a first...

Borel parts of the spectrum of an operator and of the operator algebra of a separable Hilbert space

Piotr Niemiec (2012)

Studia Mathematica

Similarity:

For a linear operator T in a Banach space let $σ_{p} (T)$ denote the point spectrum of T, let $σ_{p, n} (T)$ for finite n > 0 be the set of all $λ \in σ_{p} (T)$ such that dim ker(T - λ) = n and let $σ_{p, \infty} (T)$ be the set of all $λ \in σ_{p} (T)$ for which ker(T - λ) is infinite-dimensional. It is shown that $σ_{p} (T)$ is $ℱ_{σ}$ , $σ_{p, \infty} (T)$ is $ℱ_{σ δ}$ and for each finite n the set $σ_{p, n} (T)$ is the intersection of an $ℱ_{σ}$ set and a $_{δ}$ set provided T is closable and the domain of T is separable and weakly σ-compact. For closed densely defined operators in a separable Hilbert space a more...

Spaces with property $(D C (ω_{1}))$

Wei-Feng Xuan, Wei-Xue Shi (2017)

Commentationes Mathematicae Universitatis Carolinae

Similarity:

We prove that if $X$ is a first countable space with property $(D C (ω_{1}))$ and with a $G_{δ}$ -diagonal then the cardinality of $X$ is at most $𝔠$ . We also show that if $X$ is a first countable, DCCC, normal space then the extent of $X$ is at most $𝔠$ .

On $n$ -thin dense sets in powers of topological spaces

Adam Bartoš (2016)

Commentationes Mathematicae Universitatis Carolinae

Similarity:

A subset of a product of topological spaces is called $n$ -thin if every its two distinct points differ in at least $n$ coordinates. We generalize a construction of Gruenhage, Natkaniec, and Piotrowski, and obtain, under CH, a countable $T_{3}$ space $X$ without isolated points such that $X^{n}$ contains an $n$ -thin dense subset, but $X^{n + 1}$ does not contain any $n$ -thin dense subset. We also observe that part of the construction can be carried out under MA.

On the rigidity of webs

Michel Belliart (2007)

Bulletin de la Société Mathématique de France

Similarity:

Plane $d$ -webs have been studied a lot since their appearance at the turn of the 20th century. A rather recent and striking result for them is the theorem of Dufour, stating that the measurable conjugacies between 3-webs have to be analytic. Here, we show that even the set-theoretic conjugacies between two $d$ -webs, $d \geq 3$ are analytic unless both webs are analytically parallelizable. Between two set-theoretically conjugate parallelizable $d$ -webs, however, there always exists a nonmeasurable conjugacy;...

Functionally countable subalgebras and some properties of the Banaschewski compactification

A. R. Olfati (2016)

Commentationes Mathematicae Universitatis Carolinae

Similarity:

Let $X$ be a zero-dimensional space and $C_{c} (X)$ be the set of all continuous real valued functions on $X$ with countable image. In this article we denote by $C_{c}^{K} (X)$ (resp., $C_{c}^{ψ} (X))$ the set of all functions in $C_{c} (X)$ with compact (resp., pseudocompact) support. First, we observe that $C_{c}^{K} (X) = O_{c}^{β_{0} X ∖ X}$ (resp., $C_{c}^{ψ} (X) = M_{c}^{β_{0} X ∖ υ_{0} X}$ ), where $β_{0} X$ is the Banaschewski compactification of $X$ and $υ_{0} X$ is the $ℕ$ -compactification of $X$ . This implies that for an $ℕ$ -compact space $X$ , the intersection of all free maximal ideals in $C_{c} (X)$ is equal to $C_{c}^{K} (X)$ , i.e., $M_{c}^{β_{0} X ∖ X} = C_{c}^{K} (X)$ . By applying...

Selivanovski hard sets are hard

Janusz Pawlikowski (2015)

Fundamenta Mathematicae

Similarity:

Let $H \subseteq Z \subseteq 2^{ω}$ . For n ≥ 2, we prove that if Selivanovski measurable functions from $2^{ω}$ to Z give as preimages of H all Σₙ¹ subsets of $2^{ω}$ , then so do continuous injections.

Characterizations of $z$ -Lindelöf spaces

Ahmad Al-Omari, Takashi Noiri (2017)

Archivum Mathematicum

Similarity:

A topological space $(X, τ)$ is said to be $z$ -Lindelöf [1] if every cover of $X$ by cozero sets of $(X, τ)$ admits a countable subcover. In this paper, we obtain new characterizations and preservation theorems of $z$ -Lindelöf spaces.

Relations between Shy Sets and Sets of $ν_{p}$ -Measure Zero in Solovay’s Model

G. Pantsulaia (2004)

Bulletin of the Polish Academy of Sciences. Mathematics

Similarity:

An example of a non-zero non-atomic translation-invariant Borel measure $ν_{p}$ on the Banach space $ℓ_{p} (1 \leq p \leq \infty)$ is constructed in Solovay’s model. It is established that, for 1 ≤ p < ∞, the condition " $ν_{p}$ -almost every element of $ℓ_{p}$ has a property P" implies that “almost every” element of $ℓ_{p}$ (in the sense of [4]) has the property P. It is also shown that the converse is not valid.

On strong measure zero subsets of $^{κ} 2$

Aapo Halko, Saharon Shelah (2001)

Fundamenta Mathematicae

Similarity:

We study the generalized Cantor space $^{κ} 2$ and the generalized Baire space $^{κ} κ$ as analogues of the classical Cantor and Baire spaces. We equip $^{κ} κ$ with the topology where a basic neighborhood of a point η is the set ν: (∀j < i)(ν(j) = η(j)), where i < κ. We define the concept of a strong measure zero set of $^{κ} 2$ . We prove for successor $κ = κ^{< κ}$ that the ideal of strong measure zero sets of $^{κ} 2$ is $_{κ}$ -additive, where $_{κ}$ is the size of the smallest unbounded family in $^{κ} κ$ , and that the generalized Borel...

Cardinal invariants for κ-box products: weight, density character and Suslin number

W. W. Comfort, Ivan S. Gotchev

Similarity:

The symbol ${(X_{I})}_{κ}$ (with κ ≥ ω) denotes the space $X_{I} : = \prod_{i \in I} X_{i}$ with the κ-box topology; this has as base all sets of the form $U = \prod_{i \in I} U_{i}$ with $U_{i}$ open in $X_{i}$ and with $| i \in I : U_{i} \neq X_{i} | < κ$ . The symbols w, d and S denote respectively the weight, density character and Suslin number. Generalizing familiar classical results, the authors show inter alia: Theorem 3.1.10(b). If κ ≤ α⁺, |I| = α and each $X_{i}$ contains the discrete space 0,1 and satisfies $w (X_{i}) \leq α$ , then $w (X_{κ}) = α^{< κ}$ . Theorem 4.3.2. If $ω \leq κ \leq | I | \leq 2^{α}$ and $X = {(D (α))}^{I}$ with D(α) discrete, |D(α)| = α, then $d ({(X_{I})}_{κ}) = α^{< κ}$ . Corollaries 5.2.32(a)...

Displaying similar documents to “On the complexity of subspaces of S ω ”

Displaying similar documents to “On the complexity of subspaces of $S_{ω}$ ”