Character of points in the corona of a metric space Taras Banakh, Ostap Chervak, Lubomyr Zdomskyy Kielce-Lviv-Wien Warszawa - 2012 Taras Banakh, Ostap Chervak, Lubomyr Zdomskyy Character of points in the corona of a metric space
This topic lies in the intersection of three disciplines: Asymptotic Topology, General Topology, Set Theory. Taras Banakh, Ostap Chervak, Lubomyr Zdomskyy Character of points in the corona of a metric space
Asymptotic category Objects : Metric spaces, Morphisms : Coarse maps. A function f : X → Y between metric spaces is called coarse if ∀ δ ∈ R + ∃ ε ∈ R + ∀ x , x ′ ∈ X d X ( x , x ′ ) < δ ⇒ d Y ( f ( x ) , f ( x ′ )) < ε. Coarse maps are antipods of uniformly continuous maps. Taras Banakh, Ostap Chervak, Lubomyr Zdomskyy Character of points in the corona of a metric space
Asymptotic category Objects : Metric spaces, Morphisms : Coarse maps. A function f : X → Y between metric spaces is called coarse if ∀ δ ∈ R + ∃ ε ∈ R + ∀ x , x ′ ∈ X d X ( x , x ′ ) < δ ⇒ d Y ( f ( x ) , f ( x ′ )) < ε. Coarse maps are antipods of uniformly continuous maps. Taras Banakh, Ostap Chervak, Lubomyr Zdomskyy Character of points in the corona of a metric space
Asymptotic category Objects : Metric spaces, Morphisms : Coarse maps. A function f : X → Y between metric spaces is called coarse if ∀ δ ∈ R + ∃ ε ∈ R + ∀ x , x ′ ∈ X d X ( x , x ′ ) < δ ⇒ d Y ( f ( x ) , f ( x ′ )) < ε. Coarse maps are antipods of uniformly continuous maps. Taras Banakh, Ostap Chervak, Lubomyr Zdomskyy Character of points in the corona of a metric space
Coarse isomorphisms and coarse equivalences Def: A coarse map f : X → Y between metric spaces is called a a coarse isomorphism if f is bijective and f − 1 is coarse; a coarse equivalence if there exists a coarse map g : Y → X such that max { d X ( g ◦ f , id X ) , d Y ( f ◦ g , id Y ) } < ∞ . Example: The identity embedding Z → R is a coarse equivalence but not a coarse isomorphism. Asymptotic Topology studies properties of metric spaces preserved by coarse equivalences. Taras Banakh, Ostap Chervak, Lubomyr Zdomskyy Character of points in the corona of a metric space
Coarse isomorphisms and coarse equivalences Def: A coarse map f : X → Y between metric spaces is called a a coarse isomorphism if f is bijective and f − 1 is coarse; a coarse equivalence if there exists a coarse map g : Y → X such that max { d X ( g ◦ f , id X ) , d Y ( f ◦ g , id Y ) } < ∞ . Example: The identity embedding Z → R is a coarse equivalence but not a coarse isomorphism. Asymptotic Topology studies properties of metric spaces preserved by coarse equivalences. Taras Banakh, Ostap Chervak, Lubomyr Zdomskyy Character of points in the corona of a metric space
Coarse isomorphisms and coarse equivalences Def: A coarse map f : X → Y between metric spaces is called a a coarse isomorphism if f is bijective and f − 1 is coarse; a coarse equivalence if there exists a coarse map g : Y → X such that max { d X ( g ◦ f , id X ) , d Y ( f ◦ g , id Y ) } < ∞ . Example: The identity embedding Z → R is a coarse equivalence but not a coarse isomorphism. Asymptotic Topology studies properties of metric spaces preserved by coarse equivalences. Taras Banakh, Ostap Chervak, Lubomyr Zdomskyy Character of points in the corona of a metric space
Coarse isomorphisms and coarse equivalences Def: A coarse map f : X → Y between metric spaces is called a a coarse isomorphism if f is bijective and f − 1 is coarse; a coarse equivalence if there exists a coarse map g : Y → X such that max { d X ( g ◦ f , id X ) , d Y ( f ◦ g , id Y ) } < ∞ . Example: The identity embedding Z → R is a coarse equivalence but not a coarse isomorphism. Asymptotic Topology studies properties of metric spaces preserved by coarse equivalences. Taras Banakh, Ostap Chervak, Lubomyr Zdomskyy Character of points in the corona of a metric space
Asymptotic neighborhoods A function f : X → Y between metric spaces is bounded-to-bounded if a subset B ⊂ Y is bounded iff f − 1 ( B ) is bounded in X . Let ω ↑ X be the set of all bounded-to-bounded functions ε : X → ω . For a function ε ∈ ω ↑ X and a subset A ⊂ X let B ( A , ε ) = � B ( a , ε ( a )). a ∈ A B ( A , ε ) A Taras Banakh, Ostap Chervak, Lubomyr Zdomskyy Character of points in the corona of a metric space
Asymptotic neighborhoods A function f : X → Y between metric spaces is bounded-to-bounded if a subset B ⊂ Y is bounded iff f − 1 ( B ) is bounded in X . Let ω ↑ X be the set of all bounded-to-bounded functions ε : X → ω . For a function ε ∈ ω ↑ X and a subset A ⊂ X let B ( A , ε ) = � B ( a , ε ( a )). a ∈ A B ( A , ε ) A Taras Banakh, Ostap Chervak, Lubomyr Zdomskyy Character of points in the corona of a metric space
Asymptotic neighborhoods A function f : X → Y between metric spaces is bounded-to-bounded if a subset B ⊂ Y is bounded iff f − 1 ( B ) is bounded in X . Let ω ↑ X be the set of all bounded-to-bounded functions ε : X → ω . For a function ε ∈ ω ↑ X and a subset A ⊂ X let B ( A , ε ) = � B ( a , ε ( a )). a ∈ A B ( A , ε ) A Taras Banakh, Ostap Chervak, Lubomyr Zdomskyy Character of points in the corona of a metric space
Asymptotic neighborhoods A function f : X → Y between metric spaces is bounded-to-bounded if a subset B ⊂ Y is bounded iff f − 1 ( B ) is bounded in X . Let ω ↑ X be the set of all bounded-to-bounded functions ε : X → ω . For a function ε ∈ ω ↑ X and a subset A ⊂ X let B ( A , ε ) = � B ( a , ε ( a )). a ∈ A B ( A , ε ) A Taras Banakh, Ostap Chervak, Lubomyr Zdomskyy Character of points in the corona of a metric space
The corona of a metric space. For a metric space X let X d be X endowed with the discrete topology and β X d be the Stone-ˇ Cech compactification of X d . Let X # be the closed subset of β X d consisting of all unbounded ultrafilters. An ultrafilter F on X d is unbounded if it contains no bounded subset of X . Def: The corona of a metric space X is the quotient space X = X # / ∼ of X # by the equivalence relation identifying any ˇ ultrafilters p , q ∈ X # such that B ( P , ε ) ∩ B ( Q , ε ) � = ∅ for any P ∈ p , Q ∈ q and ε ∈ ω ↑ X . Elements of ˇ p of ultrafilters p ∈ X # . X are equivalence classes ˇ X : For any ultrafilter p ∈ X # the sets Topology of ˇ ˇ q : B ( P , ε ) ∈ q ∈ X # } , P ∈ p , ε ∈ ω ↑ X , B ( P , ε ) = { ˇ p in the corona ˇ form a base of closed neighborhoods of ˇ X . Taras Banakh, Ostap Chervak, Lubomyr Zdomskyy Character of points in the corona of a metric space
The corona of a metric space. For a metric space X let X d be X endowed with the discrete topology and β X d be the Stone-ˇ Cech compactification of X d . Let X # be the closed subset of β X d consisting of all unbounded ultrafilters. An ultrafilter F on X d is unbounded if it contains no bounded subset of X . Def: The corona of a metric space X is the quotient space X = X # / ∼ of X # by the equivalence relation identifying any ˇ ultrafilters p , q ∈ X # such that B ( P , ε ) ∩ B ( Q , ε ) � = ∅ for any P ∈ p , Q ∈ q and ε ∈ ω ↑ X . Elements of ˇ p of ultrafilters p ∈ X # . X are equivalence classes ˇ X : For any ultrafilter p ∈ X # the sets Topology of ˇ ˇ q : B ( P , ε ) ∈ q ∈ X # } , P ∈ p , ε ∈ ω ↑ X , B ( P , ε ) = { ˇ p in the corona ˇ form a base of closed neighborhoods of ˇ X . Taras Banakh, Ostap Chervak, Lubomyr Zdomskyy Character of points in the corona of a metric space
The corona of a metric space. For a metric space X let X d be X endowed with the discrete topology and β X d be the Stone-ˇ Cech compactification of X d . Let X # be the closed subset of β X d consisting of all unbounded ultrafilters. An ultrafilter F on X d is unbounded if it contains no bounded subset of X . Def: The corona of a metric space X is the quotient space X = X # / ∼ of X # by the equivalence relation identifying any ˇ ultrafilters p , q ∈ X # such that B ( P , ε ) ∩ B ( Q , ε ) � = ∅ for any P ∈ p , Q ∈ q and ε ∈ ω ↑ X . Elements of ˇ p of ultrafilters p ∈ X # . X are equivalence classes ˇ X : For any ultrafilter p ∈ X # the sets Topology of ˇ ˇ q : B ( P , ε ) ∈ q ∈ X # } , P ∈ p , ε ∈ ω ↑ X , B ( P , ε ) = { ˇ p in the corona ˇ form a base of closed neighborhoods of ˇ X . Taras Banakh, Ostap Chervak, Lubomyr Zdomskyy Character of points in the corona of a metric space
The corona of a metric space. For a metric space X let X d be X endowed with the discrete topology and β X d be the Stone-ˇ Cech compactification of X d . Let X # be the closed subset of β X d consisting of all unbounded ultrafilters. An ultrafilter F on X d is unbounded if it contains no bounded subset of X . Def: The corona of a metric space X is the quotient space X = X # / ∼ of X # by the equivalence relation identifying any ˇ ultrafilters p , q ∈ X # such that B ( P , ε ) ∩ B ( Q , ε ) � = ∅ for any P ∈ p , Q ∈ q and ε ∈ ω ↑ X . Elements of ˇ p of ultrafilters p ∈ X # . X are equivalence classes ˇ X : For any ultrafilter p ∈ X # the sets Topology of ˇ ˇ q : B ( P , ε ) ∈ q ∈ X # } , P ∈ p , ε ∈ ω ↑ X , B ( P , ε ) = { ˇ p in the corona ˇ form a base of closed neighborhoods of ˇ X . Taras Banakh, Ostap Chervak, Lubomyr Zdomskyy Character of points in the corona of a metric space
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