Some observations on filters with properties defined by open covers

Rodrigo Hernández-Gutiérrez; Paul J. Szeptycki

Displaying similar documents to “Some observations on filters with properties defined by open covers”

Guessing clubs in the generalized club filter

Bernhard König, Paul Larson, Yasuo Yoshinobu (2007)

Fundamenta Mathematicae

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We present principles for guessing clubs in the generalized club filter on $_{κ} λ$ . These principles are shown to be weaker than classical diamond principles but often serve as sufficient substitutes. One application is a new construction of a λ⁺-Suslin-tree using assumptions different from previous constructions. The other application partly solves open problems regarding the cofinality of reflection points for stationary subsets of ${[λ]}^{ℵ ₀}$ .

On meager function spaces, network character and meager convergence in topological spaces

Taras O. Banakh, Volodymyr Mykhaylyuk, Lubomyr Zdomsky (2011)

Commentationes Mathematicae Universitatis Carolinae

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For a non-isolated point $x$ of a topological space $X$ let ${nw}_{χ} (x)$ be the smallest cardinality of a family $𝒩$ of infinite subsets of $X$ such that each neighborhood $O (x) \subset X$ of $x$ contains a set $N \in 𝒩$ . We prove that (a) each infinite compact Hausdorff space $X$ contains a non-isolated point $x$ with ${nw}_{χ} (x) = ℵ_{0}$ ; (b) for each point $x \in X$ with ${nw}_{χ} (x) = ℵ_{0}$ there is an injective sequence ${(x_{n})}_{n \in ω}$ in $X$ that $ℱ$ -converges to $x$ for some meager filter $ℱ$ on $ω$ ; (c) if a functionally Hausdorff space $X$ contains an $ℱ$ -convergent injective sequence for some...

Products of topological spaces and families of filters

Paolo Lipparini (2023)

Commentationes Mathematicae Universitatis Carolinae

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We show that, under suitably general formulations, covering properties, accumulation properties and filter convergence are all equivalent notions. This general correspondence is exemplified in the study of products. We prove that a product is Lindelöf if and only if all subproducts by $\leq ω_{1}$ factors are Lindelöf. Parallel results are obtained for final $ω_{n}$ -compactness, $[λ, μ]$ -compactness, the Menger and the Rothberger properties.

Filter factors of truncated TLS regularization with multiple observations

Iveta Hnětynková, Martin Plešinger, Jana Žáková (2017)

Applications of Mathematics

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The total least squares (TLS) and truncated TLS (T-TLS) methods are widely known linear data fitting approaches, often used also in the context of very ill-conditioned, rank-deficient, or ill-posed problems. Regularization properties of T-TLS applied to linear approximation problems $A x \approx b$ were analyzed by Fierro, Golub, Hansen, and O’Leary (1997) through the so-called filter factors allowing to represent the solution in terms of a filtered pseudoinverse of $A$ applied to $b$ . This paper focuses...

The point of continuity property, neighbourhood assignments and filter convergences

Ahmed Bouziad (2012)

Fundamenta Mathematicae

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We show that for some large classes of topological spaces X and any metric space (Z,d), the point of continuity property of any function f: X → (Z,d) is equivalent to the following condition: (*) For every ε > 0, there is a neighbourhood assignment ${(V_{x})}_{x \in X}$ of X such that d(f(x),f(y)) < ε whenever $(x, y) \in V_{y} \times V_{x}$ . We also give various descriptions of the filters ℱ on the integers ℕ for which (*) is satisfied by the ℱ-limit of any sequence of continuous functions from a topological space into a metric...

Filter descriptive classes of Borel functions

Gabriel Debs, Jean Saint Raymond (2009)

Fundamenta Mathematicae

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We first prove that given any analytic filter ℱ on ω the set of all functions f on $2^{ω}$ which can be represented as the pointwise limit relative to ℱ of some sequence ${(f ₙ)}_{n \in ω}$ of continuous functions ( $f = l i m_{ℱ} f ₙ$ ), is exactly the set of all Borel functions of class ξ for some countable ordinal ξ that we call the rank of ℱ. We discuss several structural properties of this rank. For example, we prove that any free Π⁰₄ filter is of rank 1.

$P_{λ}$ -sets and skeletal mappings

Aleksander Błaszczyk, Anna Brzeska (2013)

Colloquium Mathematicae

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We prove that if the topology on the set Seq of all finite sequences of natural numbers is determined by $P_{λ}$ -filters and λ ≤ , then Seq is a $P_{λ}$ -set in its Čech-Stone compactification. This improves some results of Simon and of Juhász and Szymański. As a corollary we obtain a generalization of a result of Burke concerning skeletal maps and we partially answer a question of his.

On embedding models of arithmetic of cardinality ℵ₁ into reduced powers

Juliette Kennedy, Saharon Shelah (2003)

Fundamenta Mathematicae

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In the early 1970’s S. Tennenbaum proved that all countable models of PA₁¯ + ∀₁ -Th(ℕ) are embeddable into the reduced product $ℕ^{ω} / ℱ$ , where ℱ is the cofinite filter. In this paper we show that if M is a model of PA¯ + ∀₁ - Th(ℕ), and |M| = ℵ₁, then M is embeddable into $ℕ^{ω} / D$ , where D is any regular filter on ω.

Connected components of sets of finite perimeter and applications to image processing

Luigi Ambrosio, Vicent Caselles, Simon Masnou, Jean-Michel Morel (2001)

Journal of the European Mathematical Society

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This paper contains a systematic analysis of a natural measure theoretic notion of connectedness for sets of finite perimeter in $ℝ^{N}$ , introduced by H. Federer in the more general framework of the theory of currents. We provide a new and simpler proof of the existence and uniqueness of the decomposition into the so-called $M$ -connected components. Moreover, we study carefully the structure of the essential boundary of these components and give in particular a reconstruction formula of a set...

Distributed $H_{\infty}$ estimation for moving target under switching multi-agent network

Hu Chen, Qin Weiwei, He Bing, Liu Gang (2015)

Kybernetika

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In this paper, the distributed $H_{\infty}$ estimation problem is investigated for a moving target with local communication and switching topology. Based on the solution of the algebraic Riccati equation, a recursive algorithm is proposed using constant gain. The stability of the proposed algorithm is analysed by using the Lyapounov method, and a lower bound for estimation errors is obtained for the proposed common $H_{\infty}$ filter. Moreover, a bound for the $H_{\infty}$ parameter is obtained by means of the solution...

Topologies on groups determined by right cancellable ultrafilters

Igor V. Protasov (2009)

Commentationes Mathematicae Universitatis Carolinae

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For every discrete group $G$ , the Stone-Čech compactification $β G$ of $G$ has a natural structure of a compact right topological semigroup. An ultrafilter $p \in G^{*}$ , where $G^{*} = β G ∖ G$ , is called right cancellable if, given any $q, r \in G^{*}$ , $q p = r p$ implies $q = r$ . For every right cancellable ultrafilter $p \in G^{*}$ , we denote by $G (p)$ the group $G$ endowed with the strongest left invariant topology in which $p$ converges to the identity of $G$ . For any countable group $G$ and any right cancellable ultrafilters $p, q \in G^{*}$ , we show that $G (p)$ is homeomorphic to $G (q)$ if...

Coherent ultrafilters and nonhomogeneity

Jan Starý (2015)

Commentationes Mathematicae Universitatis Carolinae

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We introduce the notion of a coherent $P$ -ultrafilter on a complete ccc Boolean algebra, strengthening the notion of a $P$ -point on $ω$ , and show that these ultrafilters exist generically under $𝔠 = 𝔡$ . This improves the known existence result of Ketonen [On the existence of $P$ -points in the Stone-Čech compactification of integers, Fund. Math. 92 (1976), 91–94]. Similarly, the existence theorem of Canjar [On the generic existence of special ultrafilters, Proc. Amer. Math. Soc. 110 (1990), no. 1,...

On the uniform behaviour of the Frobenius closures of ideals

K. Khashyarmanesh (2007)

Colloquium Mathematicae

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Let be a proper ideal of a commutative Noetherian ring R of prime characteristic p and let Q() be the smallest positive integer m such that ${(^{F})}^{[p^{m}]} =^{[p^{m}]}$ , where $^{F}$ is the Frobenius closure of . This paper is concerned with the question whether the set $Q (^{[p^{m}]}) : m \in ℕ ₀$ is bounded. We give an affirmative answer in the case that the ideal is generated by an u.s.d-sequence c₁,..., cₙ for R such that (i) the map $R / \sum_{j = 1}^{n} R c_{j} \to R / \sum_{j = 1}^{n} R c ²_{j}$ induced by multiplication by c₁...cₙ is an R-monomorphism; (ii) for all $\in a s s (c ₁^{j}, . . ., c ₙ^{j})$ , c₁/1,..., cₙ/1 is a $R_{}$ -filter...

On preimages of ultrafilters in ZF

Horst Herrlich, Paul Howard, Kyriakos Keremedis (2016)

Commentationes Mathematicae Universitatis Carolinae

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We show that given infinite sets $X, Y$ and a function $f : X \to Y$ which is onto and $n$ -to-one for some $n \in ℕ$ , the preimage of any ultrafilter $ℱ$ of $Y$ under $f$ extends to an ultrafilter. We prove that the latter result is, in some sense, the best possible by constructing a permutation model $ℳ$ with a set of atoms $A$ and a finite-to-one onto function $f : A \to ω$ such that for each free ultrafilter of $ω$ its preimage under $f$ does not extend to an ultrafilter. In addition, we show that in $ℳ$ there exists an ultrafilter compact...