function spaces and local properties - university of oxford · function spaces and local properties...
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Function Spaces and Local Properties
Ziqin Feng and Paul Gartside
15th Topology Galway Colloquium
July 09, 2012
Ziqin Feng and Paul Gartside Function Spaces and Local Properties
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Function Spaces
Let X be a topological space.
C (X ) = {f : f : X → R which is continuous}Basic Open Set ∀f ∈ C (X ), ε > 0 and finite set F ⊂ X , define
B(f ,F , ε) = {g : |f (x)− g(x)| < ε where x ∈ F}
Function Space Cp(X ) isC (X ) with the point-wise convergence topology.
Ziqin Feng and Paul Gartside Function Spaces and Local Properties
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Function Spaces
Let X be a topological space.
C (X ) = {f : f : X → R which is continuous}Basic Open Set ∀f ∈ C (X ), ε > 0 and finite set F ⊂ X , define
B(f ,F , ε) = {g : |f (x)− g(x)| < ε where x ∈ F}
Function Space Cp(X ) isC (X ) with the point-wise convergence topology.
Ziqin Feng and Paul Gartside Function Spaces and Local Properties
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Function Spaces
Let X be a topological space.
C (X ) = {f : f : X → R which is continuous}Basic Open Set ∀f ∈ C (X ), ε > 0 and finite set F ⊂ X , define
B(f ,F , ε) = {g : |f (x)− g(x)| < ε where x ∈ F}
Function Space Cp(X ) isC (X ) with the point-wise convergence topology.
Ziqin Feng and Paul Gartside Function Spaces and Local Properties
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Generalized Metric Properties
DefinitionM1-Space is a space with a σ closure-preserving base.
DefinitionM3-Space is a space with a σ cushioned pair-base.
FactM1 implies M3.
QuestionDoes M3 imply M1?
Ziqin Feng and Paul Gartside Function Spaces and Local Properties
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Generalized Metric Properties
DefinitionM1-Space is a space with a σ closure-preserving base.
DefinitionM3-Space is a space with a σ cushioned pair-base.
FactM1 implies M3.
QuestionDoes M3 imply M1?
Ziqin Feng and Paul Gartside Function Spaces and Local Properties
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Generalized Metric Properties
DefinitionM1-Space is a space with a σ closure-preserving base.
DefinitionM3-Space is a space with a σ cushioned pair-base.
FactM1 implies M3.
QuestionDoes M3 imply M1?
Ziqin Feng and Paul Gartside Function Spaces and Local Properties
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More Definitions
I Let B be a family of subsets of X .
DefinitionB is closure-preserving if ∀B′ ⊆ B⋃
B′ =⋃{B : B ∈ B′}
I Let P be a family of pairs of subsets of X , i.e. (P1,P2) ∈ P.
DefinitionLet P is cushioned if ∀P ′ ⊆ P,⋃
{P1 : (P1,P2) ∈ P ′ ⊆⋃{P2 : (P1,P2) ∈ P ′}
I B (P) is σ-closure preserving (cushioned) if it is a countableunion of closure preserving (cushioned) collections.
Ziqin Feng and Paul Gartside Function Spaces and Local Properties
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More Definitions
I Let B be a family of subsets of X .
DefinitionB is closure-preserving if ∀B′ ⊆ B⋃
B′ =⋃{B : B ∈ B′}
I Let P be a family of pairs of subsets of X , i.e. (P1,P2) ∈ P.
DefinitionLet P is cushioned if ∀P ′ ⊆ P,⋃
{P1 : (P1,P2) ∈ P ′ ⊆⋃{P2 : (P1,P2) ∈ P ′}
I B (P) is σ-closure preserving (cushioned) if it is a countableunion of closure preserving (cushioned) collections.
Ziqin Feng and Paul Gartside Function Spaces and Local Properties
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More Definitions
I Let B be a family of subsets of X .
DefinitionB is closure-preserving if ∀B′ ⊆ B⋃
B′ =⋃{B : B ∈ B′}
I Let P be a family of pairs of subsets of X , i.e. (P1,P2) ∈ P.
DefinitionLet P is cushioned if ∀P ′ ⊆ P,⋃
{P1 : (P1,P2) ∈ P ′ ⊆⋃{P2 : (P1,P2) ∈ P ′}
I B (P) is σ-closure preserving (cushioned) if it is a countableunion of closure preserving (cushioned) collections.
Ziqin Feng and Paul Gartside Function Spaces and Local Properties
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Local Properties
DefinitionX is called a (σ−)m1 space
if X has a (σ−)closure preserving base at every x ∈ X .
DefinitionX is called a (σ−)m3 space
if X has a (σ−)cushioned pair-base at every x ∈ X .
LemmaFirst countable =⇒ m1-property =⇒ m3-property, andmi -properties =⇒ σ −mi -properties.Monotonically normal =⇒ m3-property.
Ziqin Feng and Paul Gartside Function Spaces and Local Properties
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Local Properties
DefinitionX is called a (σ−)m1 space
if X has a (σ−)closure preserving base at every x ∈ X .
DefinitionX is called a (σ−)m3 space
if X has a (σ−)cushioned pair-base at every x ∈ X .
LemmaFirst countable =⇒ m1-property =⇒ m3-property, andmi -properties =⇒ σ −mi -properties.Monotonically normal =⇒ m3-property.
Ziqin Feng and Paul Gartside Function Spaces and Local Properties
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Questions
TheoremCp(X ) is first countable ⇐⇒ X is countable.
Theorem (Gartside)
Cp(X ) is monotonically normal ⇐⇒ X is countable.
Question (Dow, Ramırez Martınez, Tkachuk)
Cp(X ) is m3 (or m1) ⇐⇒ X is countable?What if X is compact?
Ziqin Feng and Paul Gartside Function Spaces and Local Properties
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Questions
TheoremCp(X ) is first countable ⇐⇒ X is countable.
Theorem (Gartside)
Cp(X ) is monotonically normal ⇐⇒ X is countable.
Question (Dow, Ramırez Martınez, Tkachuk)
Cp(X ) is m3 (or m1) ⇐⇒ X is countable?What if X is compact?
Ziqin Feng and Paul Gartside Function Spaces and Local Properties
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Restrictions on Spaces with Cp(X ) having σ-m3-property
DefinitionA space Z is functionally countable if
every continuous f : Z → R has countable image.
TheoremIf Cp(X ) is a σ-m3 space, then X is functionally countable.
Corollary
If X is compact and Cp(X ) is a σ-m3 space, then X is scattered.
Ziqin Feng and Paul Gartside Function Spaces and Local Properties
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Restrictions on Spaces with Cp(X ) having σ-m3-property
DefinitionA space Z is functionally countable if
every continuous f : Z → R has countable image.
TheoremIf Cp(X ) is a σ-m3 space, then X is functionally countable.
Corollary
If X is compact and Cp(X ) is a σ-m3 space, then X is scattered.
Ziqin Feng and Paul Gartside Function Spaces and Local Properties
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Restrictions on Spaces with Cp(X ) having σ-m3-property
DefinitionA space Z is functionally countable if
every continuous f : Z → R has countable image.
TheoremIf Cp(X ) is a σ-m3 space, then X is functionally countable.
Corollary
If X is compact and Cp(X ) is a σ-m3 space, then X is scattered.
Ziqin Feng and Paul Gartside Function Spaces and Local Properties
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Compact Scattered Spaces
Let X be a compact scattered space with height α + 1, i.e.X (α+1) = ∅.Here X (0) be the set of isolated points of X , inductively,
I X (β) = X \ X (β−1) if β is successive;
I X (β) = X \ (⋃{X (γ) : γ < β} if β is a limit ordinal;
TheoremLet X be a compact scattered space with finite height a + 1.For each b < a, ∃Ub = {Ux : x ∈ X (≤b) and {x} = Ux ∩ X (b)}which is locally finite.
Then Cp(X ) is σ −m1.
Ziqin Feng and Paul Gartside Function Spaces and Local Properties
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Compact Scattered Spaces
Let X be a compact scattered space with height α + 1, i.e.X (α+1) = ∅.Here X (0) be the set of isolated points of X , inductively,
I X (β) = X \ X (β−1) if β is successive;
I X (β) = X \ (⋃{X (γ) : γ < β} if β is a limit ordinal;
TheoremLet X be a compact scattered space with finite height a + 1.For each b < a, ∃Ub = {Ux : x ∈ X (≤b) and {x} = Ux ∩ X (b)}which is locally finite.
Then Cp(X ) is σ −m1.
Ziqin Feng and Paul Gartside Function Spaces and Local Properties
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Compact Scattered SpacesI Let X be a compact scattered space with X (α) = {∗}I Cp(X ) is σ-m1, if
∃F =⋃Fn cofinal in ([X ]<ω,⊆) such that
∀n∀ closed neighborhood C of ∗, ∃E ∈ [X ]<ω and E ∩ C = ∅such that
∀F ∈ Fn either F ⊆ C or F ∩ E 6= ∅
U
E
Ziqin Feng and Paul Gartside Function Spaces and Local Properties
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Compact Scattered SpacesI Let X be a compact scattered space with X (α) = {∗}I Cp(X ) is σ-m1, if
∃F =⋃Fn cofinal in ([X ]<ω,⊆) such that
∀n∀ closed neighborhood C of ∗, ∃E ∈ [X ]<ω and E ∩ C = ∅such that
∀F ∈ Fn either F ⊆ C or F ∩ E 6= ∅
U
E
F
F'
Ziqin Feng and Paul Gartside Function Spaces and Local Properties
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Examples
I Cp(A(ω1)) and Cp(A(ω1)⊕ ω) are σ-m1.
I Let L(ω1) be one point Lindelofication of discrete space ω1
Cp(L(ω1)) is not σ-m1.
Ziqin Feng and Paul Gartside Function Spaces and Local Properties
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Examples
I Cp(A(ω1)) and Cp(A(ω1)⊕ ω) are σ-m1.
I Let L(ω1) be one point Lindelofication of discrete space ω1
Cp(L(ω1)) is not σ-m1.
Ziqin Feng and Paul Gartside Function Spaces and Local Properties
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Compact and Scattered SpacesSimilarly, We can prove,
I Let X be a compact scattered space with X (α) = {∗}I Cp(X ) is σ −m3, if and only if
∃F =⋃Fn cofinal in ([X ]<ω,⊆) and G : F → [X ]<ω such
that ∀n∀ closed neighborhood C , ∃E ∈ [X ]<ω and E ∩ C = ∅such that
∀F ∈ Fn either F ⊆ C or G (F ) ∩ E 6= ∅
U
E
Ziqin Feng and Paul Gartside Function Spaces and Local Properties
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Compact and Scattered SpacesSimilarly, We can prove,
I Let X be a compact scattered space with X (α) = {∗}I Cp(X ) is σ −m3, if and only if
∃F =⋃Fn cofinal in ([X ]<ω,⊆) and G : F → [X ]<ω such
that ∀n∀ closed neighborhood C , ∃E ∈ [X ]<ω and E ∩ C = ∅such that
∀F ∈ Fn either F ⊆ C or G (F ) ∩ E 6= ∅
U
E
F
F'
G(F)
Ziqin Feng and Paul Gartside Function Spaces and Local Properties
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Restrictions on Spaces with Cp(X ) having m3-property
TheoremIf Cp(X ) is a m3 space, then X does not contain an uncountableset of isolated points.
Corollary
If X is compact and Cp(X ) is a m3 space, then X is scattered andseparable.
Example
Cp(A(ω1)) is not a m3-space.
Ziqin Feng and Paul Gartside Function Spaces and Local Properties
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Restrictions on Spaces with Cp(X ) having m3-property
TheoremIf Cp(X ) is a m3 space, then X does not contain an uncountableset of isolated points.
Corollary
If X is compact and Cp(X ) is a m3 space, then X is scattered andseparable.
Example
Cp(A(ω1)) is not a m3-space.
Ziqin Feng and Paul Gartside Function Spaces and Local Properties
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Restrictions on Spaces with Cp(X ) having m3-property
TheoremIf Cp(X ) is a m3 space, then X does not contain an uncountableset of isolated points.
Corollary
If X is compact and Cp(X ) is a m3 space, then X is scattered andseparable.
Example
Cp(A(ω1)) is not a m3-space.
Ziqin Feng and Paul Gartside Function Spaces and Local Properties
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Spaces with Cp(X ) has m1 property
Let X be a compact separable scattered space with X (α) = {?}.
Theorem (†)
Cp(X ) has m1 property if
∃F cofinal in ([X ](<ω),⊆) which satisfies∀ closed neighborhood C of ?, ∃ finite set E ⊆ X \Csuch that ∀F ∈ F , E ∩ F 6= ∅ or F ⊆ C .
Ziqin Feng and Paul Gartside Function Spaces and Local Properties
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Proof of Theorem †
I Let D be the countable dense subset of X .
I Fix φ : N→ D.
I Define B = {B(0, {φ(i), 1 ≤ i ≤ |F |} ∪ F , 1/(2|F |)) : F ∈ F}.I We can verify B is a closure-preserving base of Cp(X ).
Ziqin Feng and Paul Gartside Function Spaces and Local Properties
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Proof of Theorem †
I Let D be the countable dense subset of X .
I Fix φ : N→ D.
I Define B = {B(0, {φ(i), 1 ≤ i ≤ |F |} ∪ F , 1/(2|F |)) : F ∈ F}.I We can verify B is a closure-preserving base of Cp(X ).
Ziqin Feng and Paul Gartside Function Spaces and Local Properties
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Proof of Theorem †
I Let D be the countable dense subset of X .
I Fix φ : N→ D.
I Define B = {B(0, {φ(i), 1 ≤ i ≤ |F |} ∪ F , 1/(2|F |)) : F ∈ F}.I We can verify B is a closure-preserving base of Cp(X ).
Ziqin Feng and Paul Gartside Function Spaces and Local Properties
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Proof of Theorem †
I Let D be the countable dense subset of X .
I Fix φ : N→ D.
I Define B = {B(0, {φ(i), 1 ≤ i ≤ |F |} ∪ F , 1/(2|F |)) : F ∈ F}.I We can verify B is a closure-preserving base of Cp(X ).
Ziqin Feng and Paul Gartside Function Spaces and Local Properties
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Example From Ψ Space
I Let A be a Maximal Almost Disjoint family on N;
I Let Ψ = N ∪ A.Points in N are isolated;A basic neighborhood of A ∈ A is {A} ∪ (A \ F ) where F ⊆ Nis finite;
I Let K be the one point compactifcation of Ψ. Let ? be the‘point at infinity’;
I Cp(K ) has m1 property.
This answers the question raised by Dow, Ramırez Martınezand Tkachuk.
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Example From Ψ Space
I Let A be a Maximal Almost Disjoint family on N;
I Let Ψ = N ∪ A.Points in N are isolated;A basic neighborhood of A ∈ A is {A} ∪ (A \ F ) where F ⊆ Nis finite;
I Let K be the one point compactifcation of Ψ. Let ? be the‘point at infinity’;
I Cp(K ) has m1 property.
This answers the question raised by Dow, Ramırez Martınezand Tkachuk.
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Example From Ψ Space
I Let A be a Maximal Almost Disjoint family on N;
I Let Ψ = N ∪ A.Points in N are isolated;A basic neighborhood of A ∈ A is {A} ∪ (A \ F ) where F ⊆ Nis finite;
I Let K be the one point compactifcation of Ψ. Let ? be the‘point at infinity’;
I Cp(K ) has m1 property.
This answers the question raised by Dow, Ramırez Martınezand Tkachuk.
Ziqin Feng and Paul Gartside Function Spaces and Local Properties
![Page 37: Function Spaces and Local Properties - University of Oxford · Function Spaces and Local Properties Ziqin Feng and Paul Gartside 15th Topology Galway Colloquium July 09, 2012 Ziqin](https://reader035.vdocument.in/reader035/viewer/2022071107/5fe10e6a25b4d03dc80d6c58/html5/thumbnails/37.jpg)
Example From Ψ Space
I Let A be a Maximal Almost Disjoint family on N;
I Let Ψ = N ∪ A.Points in N are isolated;A basic neighborhood of A ∈ A is {A} ∪ (A \ F ) where F ⊆ Nis finite;
I Let K be the one point compactifcation of Ψ. Let ? be the‘point at infinity’;
I Cp(K ) has m1 property.
This answers the question raised by Dow, Ramırez Martınezand Tkachuk.
Ziqin Feng and Paul Gartside Function Spaces and Local Properties
![Page 38: Function Spaces and Local Properties - University of Oxford · Function Spaces and Local Properties Ziqin Feng and Paul Gartside 15th Topology Galway Colloquium July 09, 2012 Ziqin](https://reader035.vdocument.in/reader035/viewer/2022071107/5fe10e6a25b4d03dc80d6c58/html5/thumbnails/38.jpg)
Thank You!
Ziqin Feng and Paul Gartside Function Spaces and Local Properties