Displaying similar documents to “ K σ -bounded sets and filters on ω ”

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.

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 .

A semifilter approach to selection principles II: τ * -covers

Lubomyr Zdomsky (2006)

Commentationes Mathematicae Universitatis Carolinae

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Developing the idea of assigning to a large cover of a topological space a corresponding semifilter, we show that every Menger topological space has the property ⋃ fin ( 𝒪 , T * ) provided ( 𝔲 < 𝔤 ) , and every space with the property ⋃ fin ( 𝒪 , T * ) is Hurewicz provided ( Depth + ( [ ω ] ℵ 0 ) ≤ 𝔟 ) . Combining this with the results proven in cited literature, we settle all questions whether (it is consistent that) the properties P and Q [do not] coincide, where P and Q run over ⋃ fin ( 𝒪 , Γ ) , ⋃ fin ( 𝒪 , T ) , ⋃ fin ( 𝒪 , T * ) , ⋃ fin ( 𝒪 , Ω ) , and ⋃ fin ( 𝒪 , 𝒪 ) .

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.

Seven characterizations of non-meager 𝖯-filters

Kenneth Kunen, Andrea Medini, Lyubomyr Zdomskyy (2015)

Fundamenta Mathematicae

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We give several topological/combinatorial conditions that, for a filter on ω, are equivalent to being a non-meager -filter. In particular, we show that a filter is countable dense homogeneous if and only if it is a non-meager -filter. Here, we identify a filter with a subspace of 2 ω through characteristic functions. Along the way, we generalize to non-meager -filters a result of Miller (1984) about -points, and we employ and give a new proof of results of Marciszewski (1998). We also...