Network character and filter convergence in β! Taras Banakh, V.Mykhaylyuk, L.Zdomskyy Winter School, 2011 Taras Banakh, V.Mykhaylyuk, L.Zdomskyy Network character and filter convergence in β! Filters on ! Definition A family F of non-empty subsets of ! is a filter if F is closed under intersections and taking supersets. A filter F is free if \F = ;. For a filter F on ! let F + = fA ⊂ ! : 8F 2 F A \ F 6= ;g: Basic Example The Fr´echetfilter Fr = f! n F : F is finiteg of cofinite subsets. Fr + = [!]! is the family of all infinite subsets of !. Taras Banakh, V.Mykhaylyuk, L.Zdomskyy Network character and filter convergence in β! Filters on ! Definition A family F of non-empty subsets of ! is a filter if F is closed under intersections and taking supersets. A filter F is free if \F = ;. For a filter F on ! let F + = fA ⊂ ! : 8F 2 F A \ F 6= ;g: Basic Example The Fr´echetfilter Fr = f! n F : F is finiteg of cofinite subsets. Fr + = [!]! is the family of all infinite subsets of !. Taras Banakh, V.Mykhaylyuk, L.Zdomskyy Network character and filter convergence in β! Filters on ! Definition A family F of non-empty subsets of ! is a filter if F is closed under intersections and taking supersets. A filter F is free if \F = ;. For a filter F on ! let F + = fA ⊂ ! : 8F 2 F A \ F 6= ;g: Basic Example The Fr´echetfilter Fr = f! n F : F is finiteg of cofinite subsets. Fr + = [!]! is the family of all infinite subsets of !. Taras Banakh, V.Mykhaylyuk, L.Zdomskyy Network character and filter convergence in β! Filters on ! Definition A family F of non-empty subsets of ! is a filter if F is closed under intersections and taking supersets. A filter F is free if \F = ;. For a filter F on ! let F + = fA ⊂ ! : 8F 2 F A \ F 6= ;g: Basic Example The Fr´echetfilter Fr = f! n F : F is finiteg of cofinite subsets. Fr + = [!]! is the family of all infinite subsets of !. Taras Banakh, V.Mykhaylyuk, L.Zdomskyy Network character and filter convergence in β! Filters on ! Definition A family F of non-empty subsets of ! is a filter if F is closed under intersections and taking supersets. A filter F is free if \F = ;. For a filter F on ! let F + = fA ⊂ ! : 8F 2 F A \ F 6= ;g: Basic Example The Fr´echetfilter Fr = f! n F : F is finiteg of cofinite subsets. Fr + = [!]! is the family of all infinite subsets of !. Taras Banakh, V.Mykhaylyuk, L.Zdomskyy Network character and filter convergence in β! Filter convergence Let F be a free filter on !. Definition A sequence (xn)n2! in a topological space X F-converges to a point x1 2 X if 8O(x1) 9F 2 F 8n 2 F xn 2 O(x1): Remark A sequence Fr-converges iff it converges in the standard sense. Taras Banakh, V.Mykhaylyuk, L.Zdomskyy Network character and filter convergence in β! Filter convergence Let F be a free filter on !. Definition A sequence (xn)n2! in a topological space X F-converges to a point x1 2 X if 8O(x1) 9F 2 F 8n 2 F xn 2 O(x1): Remark A sequence Fr-converges iff it converges in the standard sense. Taras Banakh, V.Mykhaylyuk, L.Zdomskyy Network character and filter convergence in β! Motivating Observation and Problem Two Motivating Facts: 1. Any sequence in a compact Hausdorff space is U-convergent for any ultrafilter U. 2. No injective sequence in β! is Fr-convergent for the Fr´echet filter Fr. Problem Which property of the Fr´echetfilter Fr is responsible for such a phenomenon? The same in other words: Problem What can be said about filters F admitting an injective F-convergent sequence in β!? We shall check three types of properties of filters: topological, measure-theoretic and combinatorial. Taras Banakh, V.Mykhaylyuk, L.Zdomskyy Network character and filter convergence in β! Motivating Observation and Problem Two Motivating Facts: 1. Any sequence in a compact Hausdorff space is U-convergent for any ultrafilter U. 2. No injective sequence in β! is Fr-convergent for the Fr´echet filter Fr. Problem Which property of the Fr´echetfilter Fr is responsible for such a phenomenon? The same in other words: Problem What can be said about filters F admitting an injective F-convergent sequence in β!? We shall check three types of properties of filters: topological, measure-theoretic and combinatorial. Taras Banakh, V.Mykhaylyuk, L.Zdomskyy Network character and filter convergence in β! Motivating Observation and Problem Two Motivating Facts: 1. Any sequence in a compact Hausdorff space is U-convergent for any ultrafilter U. 2. No injective sequence in β! is Fr-convergent for the Fr´echet filter Fr. Problem Which property of the Fr´echetfilter Fr is responsible for such a phenomenon? The same in other words: Problem What can be said about filters F admitting an injective F-convergent sequence in β!? We shall check three types of properties of filters: topological, measure-theoretic and combinatorial. Taras Banakh, V.Mykhaylyuk, L.Zdomskyy Network character and filter convergence in β! Motivating Observation and Problem Two Motivating Facts: 1. Any sequence in a compact Hausdorff space is U-convergent for any ultrafilter U. 2. No injective sequence in β! is Fr-convergent for the Fr´echet filter Fr. Problem Which property of the Fr´echetfilter Fr is responsible for such a phenomenon? The same in other words: Problem What can be said about filters F admitting an injective F-convergent sequence in β!? We shall check three types of properties of filters: topological, measure-theoretic and combinatorial. Taras Banakh, V.Mykhaylyuk, L.Zdomskyy Network character and filter convergence in β! Topological properties of filters Each filter F on ! is a subset of the power-set P(!) = 2!, which carries a nice compact metrizable topology. So, we can speak about topological properties of filters considered as subsets of the Cantor cube P(!) = 2!. Definition A filter F on ! is called meager (analytic, Fσ) if F is meager ! (analytic, Fσ) subset of 2 . It is clear that Fr ) Fσ ) analytic ) meager Taras Banakh, V.Mykhaylyuk, L.Zdomskyy Network character and filter convergence in β! Topological properties of filters Each filter F on ! is a subset of the power-set P(!) = 2!, which carries a nice compact metrizable topology. So, we can speak about topological properties of filters considered as subsets of the Cantor cube P(!) = 2!. Definition A filter F on ! is called meager (analytic, Fσ) if F is meager ! (analytic, Fσ) subset of 2 . It is clear that Fr ) Fσ ) analytic ) meager Taras Banakh, V.Mykhaylyuk, L.Zdomskyy Network character and filter convergence in β! Topological properties of filters Each filter F on ! is a subset of the power-set P(!) = 2!, which carries a nice compact metrizable topology. So, we can speak about topological properties of filters considered as subsets of the Cantor cube P(!) = 2!. Definition A filter F on ! is called meager (analytic, Fσ) if F is meager ! (analytic, Fσ) subset of 2 . It is clear that Fr ) Fσ ) analytic ) meager Taras Banakh, V.Mykhaylyuk, L.Zdomskyy Network character and filter convergence in β! Topological properties of filters Each filter F on ! is a subset of the power-set P(!) = 2!, which carries a nice compact metrizable topology. So, we can speak about topological properties of filters considered as subsets of the Cantor cube P(!) = 2!. Definition A filter F on ! is called meager (analytic, Fσ) if F is meager ! (analytic, Fσ) subset of 2 . It is clear that Fr ) Fσ ) analytic ) meager Taras Banakh, V.Mykhaylyuk, L.Zdomskyy Network character and filter convergence in β! Measure-Theoretic properties of filters The Cantor set 2! also carries a nice Borel probability measure called the Haar measure, so we can speak about measure-theoretic properties of filters. Definition A filter F on ! is measurable if it is measurable with respect to the Haar measure; null if it has Haar measure null. It is well-known that a filter is measurable if and only if it is null. Fr ) Fσ ) analytic ) meager & null Taras Banakh, V.Mykhaylyuk, L.Zdomskyy Network character and filter convergence in β! Measure-Theoretic properties of filters The Cantor set 2! also carries a nice Borel probability measure called the Haar measure, so we can speak about measure-theoretic properties of filters. Definition A filter F on ! is measurable if it is measurable with respect to the Haar measure; null if it has Haar measure null. It is well-known that a filter is measurable if and only if it is null. Fr ) Fσ ) analytic ) meager & null Taras Banakh, V.Mykhaylyuk, L.Zdomskyy Network character and filter convergence in β! Measure-Theoretic properties of filters The Cantor set 2! also carries a nice Borel probability measure called the Haar measure, so we can speak about measure-theoretic properties of filters. Definition A filter F on ! is measurable if it is measurable with respect to the Haar measure; null if it has Haar measure null. It is well-known that a filter is measurable if and only if it is null. Fr ) Fσ ) analytic ) meager & null Taras Banakh, V.Mykhaylyuk, L.Zdomskyy Network character and filter convergence in β! Measure-Theoretic properties of filters The Cantor set 2! also carries a nice Borel probability measure called the Haar measure, so we can speak about measure-theoretic properties of filters. Definition A filter F on ! is measurable if it is measurable with respect to the Haar measure; null if it has Haar measure null.