Displaying similar documents to “A semifilter approach to selection principles II: τ * -covers”

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 ) ⊂ X of x contains a set N ∈ 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 ∈ X with nw χ ( x ) = ℵ 0 there is an injective sequence ( x n ) n ∈ 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...

Some observations on filters with properties defined by open covers

Rodrigo Hernández-Gutiérrez, Paul J. Szeptycki (2015)

Commentationes Mathematicae Universitatis Carolinae

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We study the relation between the Hurewicz and Menger properties of filters considered topologically as subspaces of 𝒫 ( ω ) with the Cantor set topology.

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 ∈ X of X such that d(f(x),f(y)) < ε whenever ( x , y ) ∈ V y × 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...

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 .

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 ≤ ω 1 factors are Lindelöf. Parallel results are obtained for final ω n -compactness, [ λ , μ ] -compactness, the Menger and the Rothberger properties.

ω H-sets and cardinal invariants

Alessandro Fedeli (1998)

Commentationes Mathematicae Universitatis Carolinae

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A subset A of a Hausdorff space X is called an ω H-set in X if for every open family 𝒰 in X such that A ⊂ ⋃ 𝒰 there exists a countable subfamily 𝒱 of 𝒰 such that A ⊂ ⋃ { V ¯ : V ∈ 𝒱 } . In this paper we introduce a new cardinal function t s θ and show that | A | ≤ 2 t s θ ( X ) ψ c ( X ) for every ω H-set A of a Hausdorff space X .

On families of Lindelöf and related subspaces of 2 ω ₁

Lúcia Junqueira, Piotr Koszmider (2001)

Fundamenta Mathematicae

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We consider the families of all subspaces of size ω₁ of 2 ω ₁ (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 ₁ 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...

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 ∈ 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.