Topologically invariant -ideals on homogeneous Polish spacesset_theory/Workshop2013/... · Topologically invariant ˙-ideals on homogeneous Polish spaces Taras Banakh, Micha l Morayne,

Topologically invariant σ-idealson homogeneous Polish spaces

Taras Banakh,Micha l Morayne, Robert Ra lowski, Szymon Zeberski

Dusan Repovs

Kielce-Lviv

Warszawa - 2013

T.Banakh Topologically invariant σ-ideals on Polish spaces



Dusan Repovs

Kielce-Lviv

Warszawa - 2013




Dusan Repovs

Kielce-Lviv

Warszawa - 2013


The topic is on the border of

Set Theory & Geometric Topology.


σ-ideals on sets

Let X be a set and P(X ) be the Boolean algebra of subsets of X .

A non-empty subset I ⊂ P(X ) is ideal on X if∀A,B ∈ I ∀C ∈ P(X ) A ∪ B ∈ I and A ∩ C ∈ I.

An ideal I ⊂ P(X ) is called a σ-ideal if⋃A ∈ I for any countable subfamily A ⊂ P(X ).

Example: Trivial ideals: {∅}, [X ]≤ω, P(X ).

Def: An ideal I ⊂ P(X ) is called non-trivial ifP(X ) 6= I 6⊂ [X ]≤ω.


σ-ideals on sets







σ-ideals on sets







σ-ideals on sets







σ-ideals on sets







G -invariant ideals

Let X is a G -space(i.e., a set X endowed with a left action of a group G ).

An ideal I ⊂ P(X ) is called G -invariant if∀g ∈ G ∀A ∈ I gA ∈ I.

Example

Let X be an infinite set endowed with the natural action of thesymmetric group G = S(X ).Any G -invariant ideal I on X is equal to the ideal[X ]<κ = {A ⊂ X : |A| < κ} for some cardinal κ.


G -invariant ideals



Example



G -invariant ideals



Example



G -invariant ideals



Example



Ideals on topological spaces

An ideal I on a topological space X is called topologically invariantif I is G -invariant for the homeomorphism group G = H(X ) of X .This means that I = {h(A) : A ∈ I} for any homeomorphismh : X → X .

An ideal I on a topological space X has Borel base(resp. analytic, Fσ, Gδ, BP base) if each set A ∈ I is containrd ina set B ∈ I which is Borel (resp. analytic, Fσ, Gδ, BP) in X .

A subset B ⊂ X has the Baire Property (briefly, is a BP-set) ifthere is an open set U ⊂ X such that B4U is meager in X .












Some Classical Examples

The σ-ideal M of meager sets on each topological space X istopologically invariant and has an Fσ-base;

The σ-ideal σK generated by σ-compact subsets of atopological space X is topologically invariant and hasσ-compact base;

The σ-ideal N of Lebesgue null subsets of R has Gδ-base butis not topologically invariant.












1-generated topologically invariant ideals

Each family F of subsets of a topological space X generates thetopologically invariant ideal

σF ={

A ⊂ X : ∃(hn)n∈ω ∈ H(X )ω, (Fn)n∈ω ∈ Fω A ⊂⋃n∈ω

hn(Fn)}.

A topologically invariant σ-ideal I on X is called 1-generated ifI = σF for some family F = {F} containing a single set F ⊂ X .

Theorem

[Folklore]: The ideal M of meager subset of R is 1-generatedby the Cantor set M1

0 ⊂ R.

[Menger, 1926]: The ideal M of meager subset of Rn is1-generated by the Menger cube Mn

n−1 ⊂ Rn.

[Banakh, Repovs, 2012]: The ideal M of meager subset ofthe Hilbert cube Iω is 1-generated.




σF ={


hn(Fn)}.


Theorem


0 ⊂ R.


n−1 ⊂ Rn.





σF ={


hn(Fn)}.


Theorem


0 ⊂ R.


n−1 ⊂ Rn.





σF ={


hn(Fn)}.


Theorem


0 ⊂ R.


n−1 ⊂ Rn.





σF ={


hn(Fn)}.


Theorem


0 ⊂ R.


n−1 ⊂ Rn.



Different faces of the ideal M: the ideals σZn

A closed subset A of a topological space X is called a Zn-set forn ≤ ω if the set {f ∈ C (In,X ) : f (In) ∩ A = ∅} is dense in thefunction space C (In,X ) endowed with the compact-open topology.

Fact

A subset A ⊂ X is a Z0-set iff A is closed and nowhere dense in X .

The family Zn of all Zn-sets in X generates the topologicallyinvariant σ-ideal σZn having Fσ-base.

Fact

M = σZ0 ⊃ σZ1 ⊃ σZ2 ⊃ · · · ⊃ σZω.




Fact



Fact

M = σZ0 ⊃ σZ1 ⊃ σZ2 ⊃ · · · ⊃ σZω.




Fact



Fact

M = σZ0 ⊃ σZ1 ⊃ σZ2 ⊃ · · · ⊃ σZω.




Fact



Fact

M = σZ0 ⊃ σZ1 ⊃ σZ2 ⊃ · · · ⊃ σZω.


The ideals σDn and σDn

For a topological space X and an integer number n ∈ ω let

Dn (resp. Dn) be the family of all at most n-dimensional(closed) subsets of X ;

D<ω =⋃

n∈ω Dn, D<ω =⋃

n∈ω Dn;

σDn, σD<ω, σDn, σD<ω be the σ-ideals generated by thefamilies Dn, D<ω, Dn, D<ω.

Fact

For a metrizable separable space X the σ-ideal:

σD0 = σD<ω contains all countably-dimensional subsets of X ;

σD<ω contains all strongly countably-dimensional sets in X ;

σD0 ⊂ σD1 ⊂ σD2 ⊂ · · · ⊂ σD<ω ⊂ σD<ω = σD0.

For X = Rn, we get M = σDn−1 = σZ0.





D<ω =⋃


n∈ω Dn;


Fact




σD0 ⊂ σD1 ⊂ σD2 ⊂ · · · ⊂ σD<ω ⊂ σD<ω = σD0.






D<ω =⋃


n∈ω Dn;


Fact




σD0 ⊂ σD1 ⊂ σD2 ⊂ · · · ⊂ σD<ω ⊂ σD<ω = σD0.






D<ω =⋃


n∈ω Dn;


Fact




σD0 ⊂ σD1 ⊂ σD2 ⊂ · · · ⊂ σD<ω ⊂ σD<ω = σD0.






D<ω =⋃


n∈ω Dn;


Fact




σD0 ⊂ σD1 ⊂ σD2 ⊂ · · · ⊂ σD<ω ⊂ σD<ω = σD0.






D<ω =⋃


n∈ω Dn;


Fact




σD0 ⊂ σD1 ⊂ σD2 ⊂ · · · ⊂ σD<ω ⊂ σD<ω = σD0.






D<ω =⋃


n∈ω Dn;


Fact




σD0 ⊂ σD1 ⊂ σD2 ⊂ · · · ⊂ σD<ω ⊂ σD<ω = σD0.






D<ω =⋃


n∈ω Dn;


Fact




σD0 ⊂ σD1 ⊂ σD2 ⊂ · · · ⊂ σD<ω ⊂ σD<ω = σD0.



General Problem

Problem

Study the structure, properties, and cardinal characteristics oftopologically invariant σ-ideals with Borel base on a given Polishspace X .

In particular, evaluate the cardinal characteristics of the σ-idealsσDn, σZm and their intersections σDn ∩ σZm.


General Problem

Problem

Study the structure, properties, and cardinal characteristics oftopologically invariant σ-ideals with Borel base on a given Polishspace X .

In particular, evaluate the cardinal characteristics of the σ-idealsσDn, σZm and their intersections σDn ∩ σZm.


Topologically homogeneous spaces

A topological space X is topologically homogeneous if for anypoints x , y ∈ X there is a homeomorphism h : X → X such thath(x) = y .Examples: The spaces Rn, Iω, 2ω, ωω are topologicallyhomogeneous.

Theorem (folklore)

Each infinite zero-dimensional topologically homogeneous Polishspace X is homeomorphic to:

ω iff X is countable;

2ω iff X is uncountable and compact;

2ω × ω iff X is uncountable, locally compact, and notcompact;

ωω iff X is not locally compact.




Theorem (folklore)









Theorem (folklore)







Topologically invariant σ-ideals on topologicallyhomogeneous zero-dimensional Polish spaces

Theorem (folklore)

Each non-trivial topologically invariant σ-ideal I with analytic baseon a zero-dimensional topologically homogeneous Polish space X isequal to M or σK.

The σ-ideals M and σK are well-studied in Set Theory.


Topologically invariant σ-ideals on topologicallyhomogeneous zero-dimensional Polish spaces

Theorem (folklore)

Each non-trivial topologically invariant σ-ideal I with analytic baseon a zero-dimensional topologically homogeneous Polish space X isequal to M or σK.

The σ-ideals M and σK are well-studied in Set Theory.


Maximality of the ideal M

Theorem

For a topologically homogeneous Polish space X the ideal M ofmeager sets is:

a maximal ideal among non-trivial topologically invariantσ-ideals with BP-base on X ;

the largest ideal among non-trivial topologically invariantσ-ideals with Fσ-base on X .


Maximality of the ideal M

Theorem

For a topologically homogeneous Polish space X the ideal M ofmeager sets is:

a maximal ideal among non-trivial topologically invariantσ-ideals with BP-base on X ;

the largest ideal among non-trivial topologically invariantσ-ideals with Fσ-base on X .


The ideal M on Euclidean spaces

Theorem (Banakh, Morayne, Ra lowski, Zeberski, 2011)

Each non-trivial topologically invariant σ-ideal with BP-base on Rn

is contained in the ideal M.So, M is the largest non-trivial topologically invariant σ-ideal withBP-base on Rn.

For the Hilbert cube Iω this is not true anymore.

Example

The σ-ideal σD0 of countably-dimensional subset in the Hilbertcube Iω is non-trivial, topologically invariant, has Gδσ-base, butσD0 6⊂ M.







Example








Example








Example



The σ-ideal σC0 generated by minimal Cantor sets

A Cantor set in a topological space X is any subset C ⊂ Xhomeomorphic to the Cantor cube 2ω.A Cantor set C ⊂ X is called a minimal Cantor set in X if for eachCantor set B ⊂ X there is a homeomorphism h : X → X such thath(C ) ⊂ B.The σ-ideal σC0 generated by minimal Cantor sets in X is1-generated (by any minimal Cantor set if it exists or by ∅ if not).

Proposition

Each non-trivial topologically invariant σ-ideal I with analytic baseon a Polish space X contains the σ-ideal σC0 generated by minimalCantor sets in X .




Proposition





Proposition





Proposition



The σ-ideal σG0 generated by minimal dense Gδ-sets

A dense Gδ-set A in a Polish space X is calleda minimal dense Gδ-set in X if for each dense Gδ-set B ⊂ Xthere is a homeomorphism h : X → X such that h(A) ⊂ B.The σ-ideal σG0 generated by minimal dense Gδ-sets in X is1-generated(by any minimal dense Gδ-set if it exists or by ∅ if not).

Proposition

Let I be a topologically invariant σ-ideal I with BP-base on atopologically homogeneous Polish space X . If I 6⊂ M, then Icontains the σ-ideal σG0 generated by minimal dense Gδ-sets in X .




Proposition





Proposition



Minimality of the σ-ideals σC0 and σG0

Corollary

Let I be a topologically invariant σ-ideal with analytic base on atopologically homogeneous Polish space X .

1 If I 6⊂ [X ]≤ω, then σC0 ⊂ I;

2 If I 6⊂ M, then σG0 ⊂ I.

Problem

Given a Polish space X , study the σ-ideals σC0 and σG0 generatedby minimal Cantor sets and minimal dense Gδ-sets, respectively.


Minimality of the σ-ideals σC0 and σG0

Corollary

Let I be a topologically invariant σ-ideal with analytic base on atopologically homogeneous Polish space X .

1 If I 6⊂ [X ]≤ω, then σC0 ⊂ I;

2 If I 6⊂ M, then σG0 ⊂ I.

Problem

Given a Polish space X , study the σ-ideals σC0 and σG0 generatedby minimal Cantor sets and minimal dense Gδ-sets, respectively.


Minimal Cantor sets in Euclidean spaces

Theorem ((probably) Cantor)

Any two Cantor sets A,B ⊂ R are ambiently homeomorphic in X ,which means that h(A) = B for some homeomophism h : R→ R.

Corollary

Any Cantor set C ⊂ R is minimal.

Proposition

A Cantor set C in Rn is minimal if and only if C is tame, whichmeans that h(C ) ⊂ R× {0}n−1 for some homeomorphism h of Rn.

Thus for a Euclidean space X = Rn the ideal σC0 is not trivial.It is known that each Cantor set in Rn for n ≤ 2 is tame.A Cantor set C ⊂ Rn which is not tame is called wild.





Corollary


Proposition







Corollary


Proposition







Corollary


Proposition







Corollary


Proposition







Corollary


Proposition




A wild Cantor set in R3: the Antoine’s necklace


Z -set characterization of minimal Cantor sets in Rn.

Theorem (McMillan, 1964)

For a Cantor set C ⊂ Rn the following are equivalent:

1 C is a minimal Cantor set in Rn;

2 C is tame in Rn;

3 C is a Zk -set in Rn for all k < n;

4 C is a Zk -set in Rn for k = min{2, n − 1}.

Corollary

If X = Rn, then σC0 = σD0 ∩ σZk for k = min{2, n − 1}.


Z -set characterization of minimal Cantor sets in Rn.

Theorem (McMillan, 1964)

For a Cantor set C ⊂ Rn the following are equivalent:

1 C is a minimal Cantor set in Rn;

2 C is tame in Rn;

3 C is a Zk -set in Rn for all k < n;

4 C is a Zk -set in Rn for k = min{2, n − 1}.

Corollary

If X = Rn, then σC0 = σD0 ∩ σZk for k = min{2, n − 1}.


Z -sets in the Hilbert cube Iω.

Theorem (Unknotting Z -sets)

Any homeomorphism h : A→ B between Zω-sets in Iω can beextended to a homeomorphism h : Iω → Iω of Iω.

Theorem (Kroonenberg, 1974)

A finite-dimensional subset A ⊂ Iω is a Zω-set iff it is a Z2-set.

These two theorems imply:























Characterizing minimal Cantor sets in the Hilbert cube

Theorem

For a Cantor set C ⊂ Iω the following are equivalent:

1 C is a minimal Cantor set in Iω;

2 C is a Zω-set in Iω;

3 C is a Z2-set in Iω.

Corollary

σC0 = σD0 ∩ σZω = σD0 ∩ σZω is the smallest non-trivialtopologically invariant σ-ideal with analytic base on the Hilbertcube Iω.


Characterizing minimal Cantor sets in the Hilbert cube

Theorem

For a Cantor set C ⊂ Iω the following are equivalent:

1 C is a minimal Cantor set in Iω;

2 C is a Zω-set in Iω;

3 C is a Z2-set in Iω.

Corollary

σC0 = σD0 ∩ σZω = σD0 ∩ σZω is the smallest non-trivialtopologically invariant σ-ideal with analytic base on the Hilbertcube Iω.


Now we shall consider minimal dense Gδ-sets in Im-manifolds.

We start with minimal (dense) open sets in Im-manifolds

A (dense) open set U in a topological space X is minimal if for any(dense) open set V ⊂ X there is a homeomorphism h : X → Xsuch that h(U) ⊂ V .










Im-manifolds

Let E be a topological space. An E -manifold is a paracompacttopological space that has a cover by open sets homeomorphic toopen subspaces of the model space E .

So, for m < ω, Im-manifolds are usual m-manifolds with boundary,Iω-manifolds are Hilbert cube manifolds.


Im-manifolds




Im-manifolds




Vanishing ultrafamilies families in topological spaces

A family F of subsets of a topological space X is called vanishingif for each open cover U of X the family {F ∈ F : ∀U ∈ U F 6⊂ U}is locally finite in X .

Fact

A family F = {Fn}n∈ω of subsets of a compact metric space X isvanishing if and only if diam(Fn)→ 0 as n→∞.

A family U of subsets of a topological space X is called anultrafamily if for any distinct sets U,V ∈ Ueither U ∩ V = ∅ or U ⊂ V or V ⊂ U.




Fact






Fact




Tame balls in Im-manifolds

Let X be an Im-manifold for m ≤ ω.An open subset U ⊂ X is called a tame open ball in X if itsclosure U has an open neighborhood O(U) ⊂ X such that the pair(O(U), U) is homeomorphic to{

(Rm, Im) if m < ω

(Iω × [0,∞), Iω × [0, 1]) if m = ω.

We say that for topological spaces A ⊂ X and B ⊂ Y the pairs(A,X ) and (B,Y ) are homeomorphic if there is a homeomorphismh : X → Y such that h(A) = B.

Fact

Any tame open ball in a connected Im-manifold Xis a minimal open set in X .




(Rm, Im) if m < ω

(Iω × [0,∞), Iω × [0, 1]) if m = ω.


Fact





(Rm, Im) if m < ω

(Iω × [0,∞), Iω × [0, 1]) if m = ω.


Fact





(Rm, Im) if m < ω

(Iω × [0,∞), Iω × [0, 1]) if m = ω.


Fact



Minimal and tame open sets in Im-manifolds

An open set U ⊂ X is called tame open set in an Im-manifold X ifU =

⋃U for a vanishing family U of tame open balls with disjoint

closures in X .

Theorem (Cannon, 1973; Banakh-Repovs, 2012)

1 Each dense open set U in an Im-manifold X contains a densetame open set.

2 Any two dense tame open sets U,V in an Im-manifold X areambiently homeomorphic in X .

Corollary

Each dense tame open set in an Im-manifold Xis a minimal dense open set in X .





closures in X .




Corollary






closures in X .




Corollary






closures in X .




Corollary



Minimal dense open sets in the Square


Minimal dense open sets in Life


Minimal dense open sets in Wild Nature


Minimal and tame Gδ-sets in Im-manifolds

A subset G of an Im-manifold X is called a tame Gδ-set in X ifG =

⋃∞ U for some vanishing ultrafamily U of tame open balls inX . Here

⋃∞ U =⋂{⋃

(U \ F) : F ⊂ U , |F| <∞}.It follows that each tame Gδ-set G =

⋂n∈ω Un for some decreasing

family of tame open sets Un.

Theorem (Banakh-Repovs, 2012)

1 Any dense Gδ-subset of an Im-manifold X contains a densetame Gδ-set in X .

2 Any two dense tame Gδ-sets in an Im-manifold X areambiently homeomorphic.

Corollary

Each dense tame Gδ-set in an Im-manifold Xis a minimal dense Gδ-set in X .





⋃∞ U =⋂{⋃







Corollary






⋃∞ U =⋂{⋃







Corollary






⋃∞ U =⋂{⋃







Corollary






⋃∞ U =⋂{⋃







Corollary



Characterizing minimal dense Gδ-sets in Im-manifolds


A dense Gδ-set G in an Im-manifold X is minimal if and only ifG is tame in X .

Corollary

The ideal M of meager subsets in any Im-manifold X is1-generated (by the complement of any dense tame Gδ-set in X ).


Characterizing minimal dense Gδ-sets in Im-manifolds


A dense Gδ-set G in an Im-manifold X is minimal if and only ifG is tame in X .

Corollary

The ideal M of meager subsets in any Im-manifold X is1-generated (by the complement of any dense tame Gδ-set in X ).


Cieszelski-Wojcechowski Theorem and its corollaries

Theorem (Cieszelski, Wojcechowski, 1998)

For any dense Gδ-set G in Rn there are homeomorphismsh0, . . . , hn of Rn such that Rn =

⋃nk=0 hk(G ).

Corollary

The ideal σG0 on the space X = Rn is trivialand coincides with P(X ).


For each non-trivial topologically invariant σ-ideal I with analyticbase on an Euclidean space Rn we get σC0 ⊂ I ⊂M.





⋃nk=0 hk(G ).

Corollary








⋃nk=0 hk(G ).

Corollary





The σ-ideal σG0 on the Hilbert cube Iω

In contrast to Euclidean spaces, the σ-ideal σG0 on Iω is not trivial.

Theorem (Banakh-Morayne-Ralowski, Zeberski, 2012)

For the Hilbert cube X = IωσD<ω ⊂ σG0 ⊂ σD0 6= P(X ).

Corollary

Let I be a topologically invariant σ-ideal with BP-base on Iω.If I 6⊂ M, then σD<ω ⊂ σG0 ⊂ I.

Problem

Does σG0 contain all closed countably-dimensional subsets of Iω.Is σG0 = σD0?






Corollary


Problem







Corollary


Problem







Corollary


Problem







Corollary


Problem







Corollary


Problem



An Open Problem

So, σG0 is quite interesting extremal σ-ideal on Iω.

Problem

Calculate the cardinal characteristics of the σ-ideal σG0 on theHilbert cube Iω.


An Open Problem

So, σG0 is quite interesting extremal σ-ideal on Iω.

Problem

Calculate the cardinal characteristics of the σ-ideal σG0 on theHilbert cube Iω.


Classical cardinal characteristics of ideals

For an ideal I on a set X such that⋃I = X /∈ I consider the

following four cardinals:

add(I) = min{|A| : A ⊂ I⋃A /∈ I},

cov(I) = min{|A| : A ⊂ I,⋃A = X},

non(I) = min{|A| : A ∈ P(X ) \ I},cof(I) = min{|B| : B ⊂ I ∀A ∈ I ∃B ∈ B (A ⊂ B)}.


For any ideal I on a set⋃I = X /∈ I we get

non(I) // cof(I)

add(I) //

OO

cov(I)

OO


Cardinal characteristics for pairs of ideals

For two ideals I ⊂ J on a set X let

add(I,J ) = min{|A| : A ⊂ I⋃A /∈ J },

cof(I,J ) = min{|B| : B ⊂ J ∀A ∈ I ∃B ∈ B A ⊂ B}

Observe that

add(I) = add(I, I),cov(I) = cof(F , I),non(I) = add(F , I),cof(I) = cof(I, I),

where F is the ideal of finite subsets in X .


Cardinal characteristics for pairs of ideals

For two ideals I ⊂ J on a set X let

add(I,J ) = min{|A| : A ⊂ I⋃A /∈ J },

cof(I,J ) = min{|B| : B ⊂ J ∀A ∈ I ∃B ∈ B A ⊂ B}

Observe that

add(I) = add(I, I),cov(I) = cof(F , I),non(I) = add(F , I),cof(I) = cof(I, I),

where F is the ideal of finite subsets in X .


Relations between cardinal characteristics of ideals

For any ideals I0 ⊂ I ⊂ I1 on a set X we get:

non(I1) −→ non(I)→ non(I0)→ cof(I0, I1)→ cof(I)

6 6

add(I)→ add(I0, I1)→ cov(I1)→ cov(I) −→ cov(I0)


Cardinal characteristics of the σ-ideal σC0

Theorem (Banakh-Morayne-Ralowski-Zeberski, 2011)

If X is a topologically homogeneous Im-manifold with m ≤ ω, then

1 add(σC0) = add(σC0,M) = add(M);

2 cov(σC0) = cov(M);

3 non(σC0) = non(M);

4 cof(σC0) = cof(σC0,M) = cof(M).


Cardinal characteristics of certain σ-ideals

Corollary (Banakh, Morayne, Ra lowski, Zeberski, 2011)

Any non-trivial σ-ideal I ⊂M with analytic base on atopologically homogeneous Im-manifold X has cardinalcharacteristics:

1 add(I) ≤ add(M);

2 cov(I) = cov(M);

3 non(I) = non(M);

4 cof(I) ≥ cof(M).

non(I) non(M) // cof(M) // cof(I)

add(I) //

OO

add(M) //

OO

cov(M)

OO

cov(I)

OO


Example

The topologically invariant σ-ideal I generated by the closedinterval I× {0} on the plane R× R has

1 add(I) = ω1;

2 cov(I) = cov(M);

3 non(I) = non(M);

4 cof(I) = c.

non(I) non(M) // cof(M) // cof(I) c

ω1 add(I) //

OO

add(M) //

OO

cov(M)

OO

cov(I)

OO


Cardinal characteristics of the ideal σG0


The ideal σG0 generated by tame Gδ-sets in Iω has

1 cov(σG0) ≤ add(M),

2 non(σG0) ≥ cof(M).

non(M)→ cof(M)→ non(σG0)→ cof(σG0)

6 6

add(σG0)→ cov(σG0)→ add(M)→ cov(M)


Cardinal characteristics of “non-meager” ideals

Corollary (Banakh, Morayne, Ra lowski, Zeberski, 2012)

Any non-trivial topologically invariant σ-ideal I 6⊂ M on theHilbert cube Iω with BP-base on X has cardinal characteristics:add(I) ≤ cov(I) ≤ cov(σG0) ≤ non(σG0) ≤ non(I) ≤ cof(I).

non(σG0)→ non(I)→ cof(I)

cof(M)

6

add(M)

6

add(I)→ cov(I)→ cov(σG0)

6


The ideal σD0 of countably-dimensional sets

Example

The σ-ideal σD0 of countably-dimensional subsets in Iω has

1 add(σD0) = cov(σD0) = ω1,

2 non(σD0) = cof(σD0) = c.

non(σG0)→ non(σD0) = cof(σD0) = c

cof(M)

6

add(M)

6

ω1 = add(σD0) = cov(σD0)→ cov(σG0)

6


Cardinal characteristics in dynamics

Let I be a topologically invariant σ-ideal with analytic base on theHilbert cube X = Iω.

1) If I = [X ]≤ω, then

non(M)→ cof(M)

6 6

add(M)→ cov(M)

→non(I) cof(I) c

ω1 add(I) → cov(I)=

→

→

=




2) If I 6⊂ [X ]≤ω and I ⊂M, then

non(M)→ cof(M)

6 6

add(M)→ cov(M)

→non(I) cof(I) c

ω1 add(I) → cov(I)→

==

==

→6 6




3) If I 6⊂ M and I 6= P(X ), then

non(M)→ cof(M)

6 6

add(M)→ cov(M)

→ non(I)→ cof(I)→ c

ω1 → add(I)→ cov(I)→


Cardinal characteristics of the σ-ideals σDn ∩ σZm


If a σ-ideal I on Iω is equal to one of the σ idealsσZn, σD<m or σZn ∩ σD<m for n,m ≤ ω, then

add(I) = add(M),

cov(I) = cov(M),

non(I) = non(M),

cof(I) = cof(M).


Some Open Problems

Problem

Calculate the cardinal characteristics of the σ-ideal σG0 on Iω.Is it consistent that cov(σG0) > ω1 and non(σG0) < c?Is add(σG0) = cov(σG0) = non(σG0) = cof(σG0) = c underMartin’s Axiom or PFA?

A negative answer to this problem would follow from the positiveanswer to:

Problem

Is σG0 = σD0?

Remark

For any tame Gδ-set G ⊂ I its countable power Gω is not tame inIω. Is Gω ∈ σG0?


Some Open Problems

Problem



Problem

Is σG0 = σD0?

Remark



Some Open Problems

Problem



Problem

Is σG0 = σD0?

Remark



Some Open Problems

Problem



Problem

Is σG0 = σD0?

Remark



Some Open Problems

Problem



Problem

Is σG0 = σD0?

Remark



Some Open Problems

Problem



Problem

Is σG0 = σD0?

Remark



Another Open Problem

Theorem (Classics)

Each linear Borel subspace L in `2 is meager and hence belongs tothe ideal M = σZ0.

Theorem (Banakh, 1999; Dobrowolski, Marciszewski, 2002)

There is a linear Borel subspace L in `2 which is does not belong tothe σ-ideal σZω.

Problem

Let L be a linear Borel subspace in `2. Is L ∈ σZn for all n ∈ ω?Is L ∈ σZ1?



Theorem (Classics)




Problem




Theorem (Classics)




Problem



References

T.Banakh, M.Morayne, R.Ralowski, Sz.Zeberski,Topologically invariant σ-ideals on Euclidean spaces,preprint (http://arxiv.org/abs/1208.4823).

T.Banakh, M.Morayne, R.Ralowski, Sz.Zeberski,Topologically invariant σ-ideals on the Hilbert cube,preprint (http://arxiv.org/abs/1302.5658).

T.Banakh, D.Repovs,Universal nowhere dense subsets in locally compact manifolds,preprint (http://arxiv.org/abs/1302.5651).

T.Banakh, D.Repovs,Universal meager Fσ-sets in locally compact manifolds,to appear in CMUC (http://arxiv.org/abs/1302.5653).


References

T.Banakh, M.Morayne, R.Ralowski, Sz.Zeberski,Topologically invariant σ-ideals on Euclidean spaces,preprint (http://arxiv.org/abs/1208.4823).

T.Banakh, M.Morayne, R.Ralowski, Sz.Zeberski,Topologically invariant σ-ideals on the Hilbert cube,preprint (http://arxiv.org/abs/1302.5658).

T.Banakh, D.Repovs,Universal nowhere dense subsets in locally compact manifolds,preprint (http://arxiv.org/abs/1302.5651).

T.Banakh, D.Repovs,Universal meager Fσ-sets in locally compact manifolds,to appear in CMUC (http://arxiv.org/abs/1302.5653).


Thanks!

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Documents

Topologically invariant -ideals on homogeneous Polish spacesset_theory/Workshop2013/... · Topologically invariant ˙-ideals on homogeneous Polish spaces Taras Banakh, Micha l Morayne,