Topologically invariant -ideals on homogeneous Polish spaces Taras - - PowerPoint PPT Presentation

Topologically invariant σ-ideals

• n homogeneous Polish spaces

Taras Banakh,

Micha l Morayne, Robert Ra lowski, Szymon ˙ Zeberski Duˇ san Repovˇ s

Kielce-Lviv

Warszawa - 2013

T.Banakh Topologically invariant σ-ideals on Polish spaces

Topologically invariant σ-ideals

• n homogeneous Polish spaces

Taras Banakh,

Micha l Morayne, Robert Ra lowski, Szymon ˙ Zeberski Duˇ san Repovˇ s

Kielce-Lviv

Warszawa - 2013

T.Banakh Topologically invariant σ-ideals on Polish spaces

Topologically invariant σ-ideals

• n homogeneous Polish spaces

Taras Banakh,

Micha l Morayne, Robert Ra lowski, Szymon ˙ Zeberski Duˇ san Repovˇ s

Kielce-Lviv

Warszawa - 2013

T.Banakh Topologically invariant σ-ideals on Polish spaces

The topic is on the border of Set Theory & Geometric Topology.

T.Banakh Topologically invariant σ-ideals on Polish spaces

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 if P(X) = I ⊂ [X]≤ω.

T.Banakh Topologically invariant σ-ideals on Polish spaces

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 if P(X) = I ⊂ [X]≤ω.

T.Banakh Topologically invariant σ-ideals on Polish spaces

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 if P(X) = I ⊂ [X]≤ω.

T.Banakh Topologically invariant σ-ideals on Polish spaces

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 if P(X) = I ⊂ [X]≤ω.

T.Banakh Topologically invariant σ-ideals on Polish spaces

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 if P(X) = I ⊂ [X]≤ω.

T.Banakh Topologically invariant σ-ideals on Polish spaces

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 the symmetric 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.

T.Banakh Topologically invariant σ-ideals on Polish spaces

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 the symmetric 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.

T.Banakh Topologically invariant σ-ideals on Polish spaces

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 the symmetric 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.

T.Banakh Topologically invariant σ-ideals on Polish spaces

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 the symmetric 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.

T.Banakh Topologically invariant σ-ideals on Polish spaces

Ideals on topological spaces

An ideal I on a topological space X is called topologically invariant if I is G-invariant for the homeomorphism group G = H(X) of X. This means that I = {h(A) : A ∈ I} for any homeomorphism h : 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 in a 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) if there is an open set U ⊂ X such that B△U is meager in X.

T.Banakh Topologically invariant σ-ideals on Polish spaces

Ideals on topological spaces

An ideal I on a topological space X is called topologically invariant if I is G-invariant for the homeomorphism group G = H(X) of X. This means that I = {h(A) : A ∈ I} for any homeomorphism h : 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 in a 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) if there is an open set U ⊂ X such that B△U is meager in X.

T.Banakh Topologically invariant σ-ideals on Polish spaces

Ideals on topological spaces

An ideal I on a topological space X is called topologically invariant if I is G-invariant for the homeomorphism group G = H(X) of X. This means that I = {h(A) : A ∈ I} for any homeomorphism h : 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 in a 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) if there is an open set U ⊂ X such that B△U is meager in X.

T.Banakh Topologically invariant σ-ideals on Polish spaces

Some Classical Examples

The σ-ideal M of meager sets on each topological space X is topologically invariant and has an Fσ-base; The σ-ideal σK generated by σ-compact subsets of a topological space X is topologically invariant and has σ-compact base; The σ-ideal N of Lebesgue null subsets of R has Gδ-base but is not topologically invariant.

T.Banakh Topologically invariant σ-ideals on Polish spaces

Some Classical Examples

The σ-ideal M of meager sets on each topological space X is topologically invariant and has an Fσ-base; The σ-ideal σK generated by σ-compact subsets of a topological space X is topologically invariant and has σ-compact base; The σ-ideal N of Lebesgue null subsets of R has Gδ-base but is not topologically invariant.

T.Banakh Topologically invariant σ-ideals on Polish spaces

Some Classical Examples

The σ-ideal M of meager sets on each topological space X is topologically invariant and has an Fσ-base; The σ-ideal σK generated by σ-compact subsets of a topological space X is topologically invariant and has σ-compact base; The σ-ideal N of Lebesgue null subsets of R has Gδ-base but is not topologically invariant.

T.Banakh Topologically invariant σ-ideals on Polish spaces

1-generated topologically invariant ideals

Each family F of subsets of a topological space X generates the topologically 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 if I = σF for some family F = {F} containing a single set F ⊂ X. Theorem [Folklore]: The ideal M of meager subset of R is 1-generated by the Cantor set M1

0 ⊂ R.

[Menger, 1926]: The ideal M of meager subset of Rn is 1-generated by the Menger cube Mn

n−1 ⊂ Rn.

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

T.Banakh Topologically invariant σ-ideals on Polish spaces

1-generated topologically invariant ideals

Each family F of subsets of a topological space X generates the topologically 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 if I = σF for some family F = {F} containing a single set F ⊂ X. Theorem [Folklore]: The ideal M of meager subset of R is 1-generated by the Cantor set M1

0 ⊂ R.

[Menger, 1926]: The ideal M of meager subset of Rn is 1-generated by the Menger cube Mn

n−1 ⊂ Rn.

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

T.Banakh Topologically invariant σ-ideals on Polish spaces

1-generated topologically invariant ideals

Each family F of subsets of a topological space X generates the topologically 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 if I = σF for some family F = {F} containing a single set F ⊂ X. Theorem [Folklore]: The ideal M of meager subset of R is 1-generated by the Cantor set M1

0 ⊂ R.

[Menger, 1926]: The ideal M of meager subset of Rn is 1-generated by the Menger cube Mn

n−1 ⊂ Rn.

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

T.Banakh Topologically invariant σ-ideals on Polish spaces

1-generated topologically invariant ideals

Each family F of subsets of a topological space X generates the topologically 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 if I = σF for some family F = {F} containing a single set F ⊂ X. Theorem [Folklore]: The ideal M of meager subset of R is 1-generated by the Cantor set M1

0 ⊂ R.

[Menger, 1926]: The ideal M of meager subset of Rn is 1-generated by the Menger cube Mn

n−1 ⊂ Rn.

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

T.Banakh Topologically invariant σ-ideals on Polish spaces

1-generated topologically invariant ideals

Each family F of subsets of a topological space X generates the topologically 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 if I = σF for some family F = {F} containing a single set F ⊂ X. Theorem [Folklore]: The ideal M of meager subset of R is 1-generated by the Cantor set M1

0 ⊂ R.

[Menger, 1926]: The ideal M of meager subset of Rn is 1-generated by the Menger cube Mn

n−1 ⊂ Rn.

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

T.Banakh Topologically invariant σ-ideals on Polish spaces

Different faces of the ideal M: the ideals σZn

A closed subset A of a topological space X is called a Zn-set for n ≤ ω if the set {f ∈ C(In, X) : f (In) ∩ A = ∅} is dense in the function 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 topologically invariant σ-ideal σZn having Fσ-base. Fact M = σZ0 ⊃ σZ1 ⊃ σZ2 ⊃ · · · ⊃ σZω.

T.Banakh Topologically invariant σ-ideals on Polish spaces

Different faces of the ideal M: the ideals σZn

A closed subset A of a topological space X is called a Zn-set for n ≤ ω if the set {f ∈ C(In, X) : f (In) ∩ A = ∅} is dense in the function 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 topologically invariant σ-ideal σZn having Fσ-base. Fact M = σZ0 ⊃ σZ1 ⊃ σZ2 ⊃ · · · ⊃ σZω.

T.Banakh Topologically invariant σ-ideals on Polish spaces

Different faces of the ideal M: the ideals σZn

A closed subset A of a topological space X is called a Zn-set for n ≤ ω if the set {f ∈ C(In, X) : f (In) ∩ A = ∅} is dense in the function 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 topologically invariant σ-ideal σZn having Fσ-base. Fact M = σZ0 ⊃ σZ1 ⊃ σZ2 ⊃ · · · ⊃ σZω.

T.Banakh Topologically invariant σ-ideals on Polish spaces

Different faces of the ideal M: the ideals σZn

A closed subset A of a topological space X is called a Zn-set for n ≤ ω if the set {f ∈ C(In, X) : f (In) ∩ A = ∅} is dense in the function 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 topologically invariant σ-ideal σZn having Fσ-base. Fact M = σZ0 ⊃ σZ1 ⊃ σZ2 ⊃ · · · ⊃ σZω.

T.Banakh Topologically invariant σ-ideals on Polish spaces

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 the families 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.

T.Banakh Topologically invariant σ-ideals on Polish spaces

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 the families 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.

T.Banakh Topologically invariant σ-ideals on Polish spaces

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 the families 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.

T.Banakh Topologically invariant σ-ideals on Polish spaces

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 the families 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.

T.Banakh Topologically invariant σ-ideals on Polish spaces

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 the families 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.

T.Banakh Topologically invariant σ-ideals on Polish spaces

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 the families 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.

T.Banakh Topologically invariant σ-ideals on Polish spaces

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 the families 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.

T.Banakh Topologically invariant σ-ideals on Polish spaces

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 the families 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.

T.Banakh Topologically invariant σ-ideals on Polish spaces

General Problem

Problem Study the structure, properties, and cardinal characteristics of topologically invariant σ-ideals with Borel base on a given Polish space X. In particular, evaluate the cardinal characteristics of the σ-ideals σDn, σZm and their intersections σDn ∩ σZm.

T.Banakh Topologically invariant σ-ideals on Polish spaces

General Problem

Problem Study the structure, properties, and cardinal characteristics of topologically invariant σ-ideals with Borel base on a given Polish space X. In particular, evaluate the cardinal characteristics of the σ-ideals σDn, σZm and their intersections σDn ∩ σZm.

T.Banakh Topologically invariant σ-ideals on Polish spaces

Topologically homogeneous spaces

A topological space X is topologically homogeneous if for any points x, y ∈ X there is a homeomorphism h : X → X such that h(x) = y. Examples: The spaces Rn, Iω, 2ω, ωω are topologically homogeneous. Theorem (folklore) Each infinite zero-dimensional topologically homogeneous Polish space 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 not compact; ωω iff X is not locally compact.

T.Banakh Topologically invariant σ-ideals on Polish spaces

Topologically homogeneous spaces

A topological space X is topologically homogeneous if for any points x, y ∈ X there is a homeomorphism h : X → X such that h(x) = y. Examples: The spaces Rn, Iω, 2ω, ωω are topologically homogeneous. Theorem (folklore) Each infinite zero-dimensional topologically homogeneous Polish space 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 not compact; ωω iff X is not locally compact.

T.Banakh Topologically invariant σ-ideals on Polish spaces

Topologically homogeneous spaces

A topological space X is topologically homogeneous if for any points x, y ∈ X there is a homeomorphism h : X → X such that h(x) = y. Examples: The spaces Rn, Iω, 2ω, ωω are topologically homogeneous. Theorem (folklore) Each infinite zero-dimensional topologically homogeneous Polish space 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 not compact; ωω iff X is not locally compact.

T.Banakh Topologically invariant σ-ideals on Polish spaces

Topologically invariant σ-ideals on topologically homogeneous zero-dimensional Polish spaces

Theorem (folklore) Each non-trivial topologically invariant σ-ideal I with analytic base

• n a zero-dimensional topologically homogeneous Polish space X is

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

T.Banakh Topologically invariant σ-ideals on Polish spaces

Topologically invariant σ-ideals on topologically homogeneous zero-dimensional Polish spaces

Theorem (folklore) Each non-trivial topologically invariant σ-ideal I with analytic base

• n a zero-dimensional topologically homogeneous Polish space X is

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

T.Banakh Topologically invariant σ-ideals on Polish spaces

Maximality of the ideal M

Theorem For a topologically homogeneous Polish space X the ideal M of meager sets is: a maximal ideal among non-trivial topologically invariant σ-ideals with BP-base on X; the largest ideal among non-trivial topologically invariant σ-ideals with Fσ-base on X.

T.Banakh Topologically invariant σ-ideals on Polish spaces

Maximality of the ideal M

Theorem For a topologically homogeneous Polish space X the ideal M of meager sets is: a maximal ideal among non-trivial topologically invariant σ-ideals with BP-base on X; the largest ideal among non-trivial topologically invariant σ-ideals with Fσ-base on X.

T.Banakh Topologically invariant σ-ideals on Polish spaces

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 with BP-base on Rn. For the Hilbert cube Iω this is not true anymore. Example The σ-ideal σD0 of countably-dimensional subset in the Hilbert cube Iω is non-trivial, topologically invariant, has Gδσ-base, but σD0 ⊂ M.

T.Banakh Topologically invariant σ-ideals on Polish spaces

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 with BP-base on Rn. For the Hilbert cube Iω this is not true anymore. Example The σ-ideal σD0 of countably-dimensional subset in the Hilbert cube Iω is non-trivial, topologically invariant, has Gδσ-base, but σD0 ⊂ M.

T.Banakh Topologically invariant σ-ideals on Polish spaces

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 with BP-base on Rn. For the Hilbert cube Iω this is not true anymore. Example The σ-ideal σD0 of countably-dimensional subset in the Hilbert cube Iω is non-trivial, topologically invariant, has Gδσ-base, but σD0 ⊂ M.

T.Banakh Topologically invariant σ-ideals on Polish spaces

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 with BP-base on Rn. For the Hilbert cube Iω this is not true anymore. Example The σ-ideal σD0 of countably-dimensional subset in the Hilbert cube Iω is non-trivial, topologically invariant, has Gδσ-base, but σD0 ⊂ M.

T.Banakh Topologically invariant σ-ideals on Polish spaces

The σ-ideal σC0 generated by minimal Cantor sets

A Cantor set in a topological space X is any subset C ⊂ X homeomorphic to the Cantor cube 2ω. A Cantor set C ⊂ X is called a minimal Cantor set in X if for each Cantor set B ⊂ X there is a homeomorphism h : X → X such that h(C) ⊂ B. The σ-ideal σC0 generated by minimal Cantor sets in X is 1-generated (by any minimal Cantor set if it exists or by ∅ if not). Proposition Each non-trivial topologically invariant σ-ideal I with analytic base

• n a Polish space X contains the σ-ideal σC0 generated by minimal

Cantor sets in X.

T.Banakh Topologically invariant σ-ideals on Polish spaces

The σ-ideal σC0 generated by minimal Cantor sets

A Cantor set in a topological space X is any subset C ⊂ X homeomorphic to the Cantor cube 2ω. A Cantor set C ⊂ X is called a minimal Cantor set in X if for each Cantor set B ⊂ X there is a homeomorphism h : X → X such that h(C) ⊂ B. The σ-ideal σC0 generated by minimal Cantor sets in X is 1-generated (by any minimal Cantor set if it exists or by ∅ if not). Proposition Each non-trivial topologically invariant σ-ideal I with analytic base

• n a Polish space X contains the σ-ideal σC0 generated by minimal

Cantor sets in X.

T.Banakh Topologically invariant σ-ideals on Polish spaces

The σ-ideal σC0 generated by minimal Cantor sets

A Cantor set in a topological space X is any subset C ⊂ X homeomorphic to the Cantor cube 2ω. A Cantor set C ⊂ X is called a minimal Cantor set in X if for each Cantor set B ⊂ X there is a homeomorphism h : X → X such that h(C) ⊂ B. The σ-ideal σC0 generated by minimal Cantor sets in X is 1-generated (by any minimal Cantor set if it exists or by ∅ if not). Proposition Each non-trivial topologically invariant σ-ideal I with analytic base

• n a Polish space X contains the σ-ideal σC0 generated by minimal

Cantor sets in X.

T.Banakh Topologically invariant σ-ideals on Polish spaces

The σ-ideal σC0 generated by minimal Cantor sets

A Cantor set in a topological space X is any subset C ⊂ X homeomorphic to the Cantor cube 2ω. A Cantor set C ⊂ X is called a minimal Cantor set in X if for each Cantor set B ⊂ X there is a homeomorphism h : X → X such that h(C) ⊂ B. The σ-ideal σC0 generated by minimal Cantor sets in X is 1-generated (by any minimal Cantor set if it exists or by ∅ if not). Proposition Each non-trivial topologically invariant σ-ideal I with analytic base

• n a Polish space X contains the σ-ideal σC0 generated by minimal

Cantor sets in X.

T.Banakh Topologically invariant σ-ideals on Polish spaces

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

A dense Gδ-set A in a Polish space X is called a minimal dense Gδ-set in X if for each dense Gδ-set B ⊂ X there is a homeomorphism h : X → X such that h(A) ⊂ B. The σ-ideal σG0 generated by minimal dense Gδ-sets in X is 1-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 a topologically homogeneous Polish space X. If I ⊂ M, then I contains the σ-ideal σG0 generated by minimal dense Gδ-sets in X.

T.Banakh Topologically invariant σ-ideals on Polish spaces

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

A dense Gδ-set A in a Polish space X is called a minimal dense Gδ-set in X if for each dense Gδ-set B ⊂ X there is a homeomorphism h : X → X such that h(A) ⊂ B. The σ-ideal σG0 generated by minimal dense Gδ-sets in X is 1-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 a topologically homogeneous Polish space X. If I ⊂ M, then I contains the σ-ideal σG0 generated by minimal dense Gδ-sets in X.

T.Banakh Topologically invariant σ-ideals on Polish spaces

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

A dense Gδ-set A in a Polish space X is called a minimal dense Gδ-set in X if for each dense Gδ-set B ⊂ X there is a homeomorphism h : X → X such that h(A) ⊂ B. The σ-ideal σG0 generated by minimal dense Gδ-sets in X is 1-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 a topologically homogeneous Polish space X. If I ⊂ M, then I contains the σ-ideal σG0 generated by minimal dense Gδ-sets in X.

T.Banakh Topologically invariant σ-ideals on Polish spaces

Minimality of the σ-ideals σC0 and σG0

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

1 If I ⊂ [X]≤ω, then σC0 ⊂ I; 2 If I ⊂ M, then σG0 ⊂ I.

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

T.Banakh Topologically invariant σ-ideals on Polish spaces

Minimality of the σ-ideals σC0 and σG0

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

1 If I ⊂ [X]≤ω, then σC0 ⊂ I; 2 If I ⊂ M, then σG0 ⊂ I.

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

T.Banakh Topologically invariant σ-ideals on Polish spaces

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, which means 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.

T.Banakh Topologically invariant σ-ideals on Polish spaces

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, which means 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.

T.Banakh Topologically invariant σ-ideals on Polish spaces

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, which means 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.

T.Banakh Topologically invariant σ-ideals on Polish spaces

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, which means 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.

T.Banakh Topologically invariant σ-ideals on Polish spaces

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, which means 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.

T.Banakh Topologically invariant σ-ideals on Polish spaces

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, which means 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.

T.Banakh Topologically invariant σ-ideals on Polish spaces

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

T.Banakh Topologically invariant σ-ideals on Polish spaces

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}.

T.Banakh Topologically invariant σ-ideals on Polish spaces

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}.

T.Banakh Topologically invariant σ-ideals on Polish spaces

Z-sets in the Hilbert cube Iω.

Theorem (Unknotting Z-sets) Any homeomorphism h : A → B between Zω-sets in Iω can be extended 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:

T.Banakh Topologically invariant σ-ideals on Polish spaces

Z-sets in the Hilbert cube Iω.

Theorem (Unknotting Z-sets) Any homeomorphism h : A → B between Zω-sets in Iω can be extended 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:

T.Banakh Topologically invariant σ-ideals on Polish spaces

Z-sets in the Hilbert cube Iω.

Theorem (Unknotting Z-sets) Any homeomorphism h : A → B between Zω-sets in Iω can be extended 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:

T.Banakh Topologically invariant σ-ideals on Polish spaces

Z-sets in the Hilbert cube Iω.

Theorem (Unknotting Z-sets) Any homeomorphism h : A → B between Zω-sets in Iω can be extended 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:

T.Banakh Topologically invariant σ-ideals on Polish spaces

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-trivial topologically invariant σ-ideal with analytic base on the Hilbert cube Iω.

T.Banakh Topologically invariant σ-ideals on Polish spaces

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-trivial topologically invariant σ-ideal with analytic base on the Hilbert cube Iω.

T.Banakh Topologically invariant σ-ideals on Polish spaces

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 → X such that h(U) ⊂ V .

T.Banakh Topologically invariant σ-ideals on Polish spaces

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 → X such that h(U) ⊂ V .

T.Banakh Topologically invariant σ-ideals on Polish spaces

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 → X such that h(U) ⊂ V .

T.Banakh Topologically invariant σ-ideals on Polish spaces

Im-manifolds

Let E be a topological space. An E-manifold is a paracompact topological space that has a cover by open sets homeomorphic to

• pen subspaces of the model space E.

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

T.Banakh Topologically invariant σ-ideals on Polish spaces

Im-manifolds

Let E be a topological space. An E-manifold is a paracompact topological space that has a cover by open sets homeomorphic to

• pen subspaces of the model space E.

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

T.Banakh Topologically invariant σ-ideals on Polish spaces

Im-manifolds

Let E be a topological space. An E-manifold is a paracompact topological space that has a cover by open sets homeomorphic to

• pen subspaces of the model space E.

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

T.Banakh Topologically invariant σ-ideals on Polish spaces

Vanishing ultrafamilies families in topological spaces

A family F of subsets of a topological space X is called vanishing if for each open cover U of X the family {F ∈ F : ∀U ∈ U F ⊂ U} is locally finite in X. Fact A family F = {Fn}n∈ω of subsets of a compact metric space X is vanishing if and only if diam(Fn) → 0 as n → ∞. A family U of subsets of a topological space X is called an ultrafamily if for any distinct sets U, V ∈ U either ¯ U ∩ ¯ V = ∅ or ¯ U ⊂ V or ¯ V ⊂ U.

T.Banakh Topologically invariant σ-ideals on Polish spaces

Vanishing ultrafamilies families in topological spaces

A family F of subsets of a topological space X is called vanishing if for each open cover U of X the family {F ∈ F : ∀U ∈ U F ⊂ U} is locally finite in X. Fact A family F = {Fn}n∈ω of subsets of a compact metric space X is vanishing if and only if diam(Fn) → 0 as n → ∞. A family U of subsets of a topological space X is called an ultrafamily if for any distinct sets U, V ∈ U either ¯ U ∩ ¯ V = ∅ or ¯ U ⊂ V or ¯ V ⊂ U.

T.Banakh Topologically invariant σ-ideals on Polish spaces

Vanishing ultrafamilies families in topological spaces

A family F of subsets of a topological space X is called vanishing if for each open cover U of X the family {F ∈ F : ∀U ∈ U F ⊂ U} is locally finite in X. Fact A family F = {Fn}n∈ω of subsets of a compact metric space X is vanishing if and only if diam(Fn) → 0 as n → ∞. A family U of subsets of a topological space X is called an ultrafamily if for any distinct sets U, V ∈ U either ¯ U ∩ ¯ V = ∅ or ¯ U ⊂ V or ¯ V ⊂ U.

T.Banakh Topologically invariant σ-ideals on Polish spaces

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 its closure 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 homeomorphism h : X → Y such that h(A) = B. Fact Any tame open ball in a connected Im-manifold X is a minimal open set in X.

T.Banakh Topologically invariant σ-ideals on Polish spaces

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 its closure 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 homeomorphism h : X → Y such that h(A) = B. Fact Any tame open ball in a connected Im-manifold X is a minimal open set in X.

T.Banakh Topologically invariant σ-ideals on Polish spaces

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 its closure 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 homeomorphism h : X → Y such that h(A) = B. Fact Any tame open ball in a connected Im-manifold X is a minimal open set in X.

T.Banakh Topologically invariant σ-ideals on Polish spaces

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 its closure 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 homeomorphism h : X → Y such that h(A) = B. Fact Any tame open ball in a connected Im-manifold X is a minimal open set in X.

T.Banakh Topologically invariant σ-ideals on Polish spaces

Minimal and tame open sets in Im-manifolds

An open set U ⊂ X is called tame open set in an Im-manifold X if U = U for a vanishing family U of tame open balls with disjoint closures in X. Theorem (Cannon, 1973; Banakh-Repovˇ s, 2012)

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

tame open set.

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

ambiently homeomorphic in X. Corollary Each dense tame open set in an Im-manifold X is a minimal dense open set in X.

T.Banakh Topologically invariant σ-ideals on Polish spaces

Minimal and tame open sets in Im-manifolds

An open set U ⊂ X is called tame open set in an Im-manifold X if U = U for a vanishing family U of tame open balls with disjoint closures in X. Theorem (Cannon, 1973; Banakh-Repovˇ s, 2012)

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

tame open set.

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

ambiently homeomorphic in X. Corollary Each dense tame open set in an Im-manifold X is a minimal dense open set in X.

T.Banakh Topologically invariant σ-ideals on Polish spaces

Minimal and tame open sets in Im-manifolds

An open set U ⊂ X is called tame open set in an Im-manifold X if U = U for a vanishing family U of tame open balls with disjoint closures in X. Theorem (Cannon, 1973; Banakh-Repovˇ s, 2012)

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

tame open set.

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

ambiently homeomorphic in X. Corollary Each dense tame open set in an Im-manifold X is a minimal dense open set in X.

T.Banakh Topologically invariant σ-ideals on Polish spaces

Minimal and tame open sets in Im-manifolds

An open set U ⊂ X is called tame open set in an Im-manifold X if U = U for a vanishing family U of tame open balls with disjoint closures in X. Theorem (Cannon, 1973; Banakh-Repovˇ s, 2012)

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

tame open set.

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

ambiently homeomorphic in X. Corollary Each dense tame open set in an Im-manifold X is a minimal dense open set in X.

T.Banakh Topologically invariant σ-ideals on Polish spaces

Minimal dense open sets in the Square

T.Banakh Topologically invariant σ-ideals on Polish spaces

Minimal dense open sets in Life

T.Banakh Topologically invariant σ-ideals on Polish spaces

Minimal dense open sets in Wild Nature

T.Banakh Topologically invariant σ-ideals on Polish spaces

Minimal and tame Gδ-sets in Im-manifolds

A subset G of an Im-manifold X is called a tame Gδ-set in X if G = ∞ U for some vanishing ultrafamily U of tame open balls in

• X. 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-Repovˇ s, 2012)

1 Any dense Gδ-subset of an Im-manifold X contains a dense

tame Gδ-set in X.

2 Any two dense tame Gδ-sets in an Im-manifold X are

ambiently homeomorphic. Corollary Each dense tame Gδ-set in an Im-manifold X is a minimal dense Gδ-set in X.

T.Banakh Topologically invariant σ-ideals on Polish spaces

Minimal and tame Gδ-sets in Im-manifolds

A subset G of an Im-manifold X is called a tame Gδ-set in X if G = ∞ U for some vanishing ultrafamily U of tame open balls in

• X. 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-Repovˇ s, 2012)

1 Any dense Gδ-subset of an Im-manifold X contains a dense

tame Gδ-set in X.

2 Any two dense tame Gδ-sets in an Im-manifold X are

ambiently homeomorphic. Corollary Each dense tame Gδ-set in an Im-manifold X is a minimal dense Gδ-set in X.

T.Banakh Topologically invariant σ-ideals on Polish spaces

Minimal and tame Gδ-sets in Im-manifolds

A subset G of an Im-manifold X is called a tame Gδ-set in X if G = ∞ U for some vanishing ultrafamily U of tame open balls in

• X. 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-Repovˇ s, 2012)

1 Any dense Gδ-subset of an Im-manifold X contains a dense

tame Gδ-set in X.

2 Any two dense tame Gδ-sets in an Im-manifold X are

ambiently homeomorphic. Corollary Each dense tame Gδ-set in an Im-manifold X is a minimal dense Gδ-set in X.

T.Banakh Topologically invariant σ-ideals on Polish spaces

Minimal and tame Gδ-sets in Im-manifolds

A subset G of an Im-manifold X is called a tame Gδ-set in X if G = ∞ U for some vanishing ultrafamily U of tame open balls in

• X. 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-Repovˇ s, 2012)

1 Any dense Gδ-subset of an Im-manifold X contains a dense

tame Gδ-set in X.

2 Any two dense tame Gδ-sets in an Im-manifold X are

ambiently homeomorphic. Corollary Each dense tame Gδ-set in an Im-manifold X is a minimal dense Gδ-set in X.

T.Banakh Topologically invariant σ-ideals on Polish spaces

Minimal and tame Gδ-sets in Im-manifolds

A subset G of an Im-manifold X is called a tame Gδ-set in X if G = ∞ U for some vanishing ultrafamily U of tame open balls in

• X. 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-Repovˇ s, 2012)

1 Any dense Gδ-subset of an Im-manifold X contains a dense

tame Gδ-set in X.

2 Any two dense tame Gδ-sets in an Im-manifold X are

ambiently homeomorphic. Corollary Each dense tame Gδ-set in an Im-manifold X is a minimal dense Gδ-set in X.

T.Banakh Topologically invariant σ-ideals on Polish spaces

Characterizing minimal dense Gδ-sets in Im-manifolds

Theorem (Banakh-Repovˇ s, 2012) A dense Gδ-set G in an Im-manifold X is minimal if and only if G is tame in X. Corollary The ideal M of meager subsets in any Im-manifold X is 1-generated (by the complement of any dense tame Gδ-set in X).

T.Banakh Topologically invariant σ-ideals on Polish spaces

Characterizing minimal dense Gδ-sets in Im-manifolds

Theorem (Banakh-Repovˇ s, 2012) A dense Gδ-set G in an Im-manifold X is minimal if and only if G is tame in X. Corollary The ideal M of meager subsets in any Im-manifold X is 1-generated (by the complement of any dense tame Gδ-set in X).

T.Banakh Topologically invariant σ-ideals on Polish spaces

Cieszelski-Wojcechowski Theorem and its corollaries

Theorem (Cieszelski, Wojcechowski, 1998) For any dense Gδ-set G in Rn there are homeomorphisms h0, . . . , hn of Rn such that Rn = n

k=0 hk(G).

Corollary The ideal σG0 on the space X = Rn is trivial and coincides with P(X). Theorem (Banakh, Morayne, Ra lowski, ˙ Zeberski, 2011) For each non-trivial topologically invariant σ-ideal I with analytic base on an Euclidean space Rn we get σC0 ⊂ I ⊂ M.

T.Banakh Topologically invariant σ-ideals on Polish spaces

Cieszelski-Wojcechowski Theorem and its corollaries

Theorem (Cieszelski, Wojcechowski, 1998) For any dense Gδ-set G in Rn there are homeomorphisms h0, . . . , hn of Rn such that Rn = n

k=0 hk(G).

Corollary The ideal σG0 on the space X = Rn is trivial and coincides with P(X). Theorem (Banakh, Morayne, Ra lowski, ˙ Zeberski, 2011) For each non-trivial topologically invariant σ-ideal I with analytic base on an Euclidean space Rn we get σC0 ⊂ I ⊂ M.

T.Banakh Topologically invariant σ-ideals on Polish spaces

Cieszelski-Wojcechowski Theorem and its corollaries

Theorem (Cieszelski, Wojcechowski, 1998) For any dense Gδ-set G in Rn there are homeomorphisms h0, . . . , hn of Rn such that Rn = n

k=0 hk(G).

Corollary The ideal σG0 on the space X = Rn is trivial and coincides with P(X). Theorem (Banakh, Morayne, Ra lowski, ˙ Zeberski, 2011) For each non-trivial topologically invariant σ-ideal I with analytic base on an Euclidean space Rn we get σC0 ⊂ I ⊂ M.

T.Banakh Topologically invariant σ-ideals on Polish spaces

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 = Iω σD<ω ⊂ σG0 ⊂ σD0 = P(X). Corollary Let I be a topologically invariant σ-ideal with BP-base on Iω. If I ⊂ M, then σD<ω ⊂ σG0 ⊂ I. Problem Does σG0 contain all closed countably-dimensional subsets of Iω. Is σG0 = σD0?

T.Banakh Topologically invariant σ-ideals on Polish spaces

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 = Iω σD<ω ⊂ σG0 ⊂ σD0 = P(X). Corollary Let I be a topologically invariant σ-ideal with BP-base on Iω. If I ⊂ M, then σD<ω ⊂ σG0 ⊂ I. Problem Does σG0 contain all closed countably-dimensional subsets of Iω. Is σG0 = σD0?

T.Banakh Topologically invariant σ-ideals on Polish spaces

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 = Iω σD<ω ⊂ σG0 ⊂ σD0 = P(X). Corollary Let I be a topologically invariant σ-ideal with BP-base on Iω. If I ⊂ M, then σD<ω ⊂ σG0 ⊂ I. Problem Does σG0 contain all closed countably-dimensional subsets of Iω. Is σG0 = σD0?

T.Banakh Topologically invariant σ-ideals on Polish spaces

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 = Iω σD<ω ⊂ σG0 ⊂ σD0 = P(X). Corollary Let I be a topologically invariant σ-ideal with BP-base on Iω. If I ⊂ M, then σD<ω ⊂ σG0 ⊂ I. Problem Does σG0 contain all closed countably-dimensional subsets of Iω. Is σG0 = σD0?

T.Banakh Topologically invariant σ-ideals on Polish spaces

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 = Iω σD<ω ⊂ σG0 ⊂ σD0 = P(X). Corollary Let I be a topologically invariant σ-ideal with BP-base on Iω. If I ⊂ M, then σD<ω ⊂ σG0 ⊂ I. Problem Does σG0 contain all closed countably-dimensional subsets of Iω. Is σG0 = σD0?

T.Banakh Topologically invariant σ-ideals on Polish spaces

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 = Iω σD<ω ⊂ σG0 ⊂ σD0 = P(X). Corollary Let I be a topologically invariant σ-ideal with BP-base on Iω. If I ⊂ M, then σD<ω ⊂ σG0 ⊂ I. Problem Does σG0 contain all closed countably-dimensional subsets of Iω. Is σG0 = σD0?

T.Banakh Topologically invariant σ-ideals on Polish spaces

An Open Problem

So, σG0 is quite interesting extremal σ-ideal on Iω. Problem Calculate the cardinal characteristics of the σ-ideal σG0 on the Hilbert cube Iω.

T.Banakh Topologically invariant σ-ideals on Polish spaces

An Open Problem

So, σG0 is quite interesting extremal σ-ideal on Iω. Problem Calculate the cardinal characteristics of the σ-ideal σG0 on the Hilbert cube Iω.

T.Banakh Topologically invariant σ-ideals on Polish spaces

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)}.

T.Banakh Topologically invariant σ-ideals on Polish spaces

For any ideal I on a set I = X / ∈ I we get non(I)

cof(I)

add(I)

• cov(I)
• T.Banakh

Topologically invariant σ-ideals on Polish spaces

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.

T.Banakh Topologically invariant σ-ideals on Polish spaces

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.

T.Banakh Topologically invariant σ-ideals on Polish spaces

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

T.Banakh Topologically invariant σ-ideals on Polish spaces

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). T.Banakh Topologically invariant σ-ideals on Polish spaces

Cardinal characteristics of certain σ-ideals

Corollary (Banakh, Morayne, Ra lowski, ˙ Zeberski, 2011) Any non-trivial σ-ideal I ⊂ M with analytic base on a topologically homogeneous Im-manifold X has cardinal characteristics:

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)

• add(M)
• cov(M)
• cov(I)
• T.Banakh

Topologically invariant σ-ideals on Polish spaces

Example The topologically invariant σ-ideal I generated by the closed interval 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)

• add(M)
• cov(M)
• cov(I)
• T.Banakh

Topologically invariant σ-ideals on Polish spaces

Cardinal characteristics of the ideal σG0

Theorem (Banakh, Morayne, Ra lowski, ˙ Zeberski, 2012) 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) ✻ ✻ add(σG0) → cov(σG0) → add(M) → cov(M)

T.Banakh Topologically invariant σ-ideals on Polish spaces

Cardinal characteristics of “non-meager” ideals

Corollary (Banakh, Morayne, Ra lowski, ˙ Zeberski, 2012) Any non-trivial topologically invariant σ-ideal I ⊂ M on the Hilbert 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)

✻

add(M)

✻

add(I) → cov(I) → cov(σG0)

✻

T.Banakh Topologically invariant σ-ideals on Polish spaces

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) ✻ add(M) ✻ ω1 = add(σD0) = cov(σD0) → cov(σG0) ✻

T.Banakh Topologically invariant σ-ideals on Polish spaces

Cardinal characteristics in dynamics

Let I be a topologically invariant σ-ideal with analytic base on the Hilbert cube X = Iω. 1) If I = [X]≤ω, then

non(M) → cof(M) ✻ ✻ add(M) → cov(M) → non(I) cof(I) c ω1 add(I) → cov(I) = → → =

T.Banakh Topologically invariant σ-ideals on Polish spaces

Cardinal characteristics in dynamics

Let I be a topologically invariant σ-ideal with analytic base on the Hilbert cube X = Iω. 2) If I ⊂ [X]≤ω and I ⊂ M, then

non(M) → cof(M) ✻ ✻ add(M) → cov(M) → non(I) cof(I) c ω1 add(I) → cov(I) → = = = = → ✻ ✻

T.Banakh Topologically invariant σ-ideals on Polish spaces

Cardinal characteristics in dynamics

Let I be a topologically invariant σ-ideal with analytic base on the Hilbert cube X = Iω. 3) If I ⊂ M and I = P(X), then

non(M) → cof(M) ✻ ✻ add(M) → cov(M) → non(I) → cof(I) → c ω1 → add(I) → cov(I) →

T.Banakh Topologically invariant σ-ideals on Polish spaces

Cardinal characteristics of the σ-ideals σDn ∩ σZm

Theorem (Banakh, Morayne, Ra lowski, ˙ Zeberski, 2012) If a σ-ideal I on Iω is equal to one of the σ ideals σZn, 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).

T.Banakh Topologically invariant σ-ideals on Polish spaces

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 under Martin’s Axiom or PFA? A negative answer to this problem would follow from the positive answer to: Problem Is σG0 = σD0? Remark For any tame Gδ-set G ⊂ I its countable power G ω is not tame in Iω. Is G ω ∈ σG0?

T.Banakh Topologically invariant σ-ideals on Polish spaces

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 under Martin’s Axiom or PFA? A negative answer to this problem would follow from the positive answer to: Problem Is σG0 = σD0? Remark For any tame Gδ-set G ⊂ I its countable power G ω is not tame in Iω. Is G ω ∈ σG0?

T.Banakh Topologically invariant σ-ideals on Polish spaces

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 under Martin’s Axiom or PFA? A negative answer to this problem would follow from the positive answer to: Problem Is σG0 = σD0? Remark For any tame Gδ-set G ⊂ I its countable power G ω is not tame in Iω. Is G ω ∈ σG0?

T.Banakh Topologically invariant σ-ideals on Polish spaces

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 under Martin’s Axiom or PFA? A negative answer to this problem would follow from the positive answer to: Problem Is σG0 = σD0? Remark For any tame Gδ-set G ⊂ I its countable power G ω is not tame in Iω. Is G ω ∈ σG0?

T.Banakh Topologically invariant σ-ideals on Polish spaces

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 under Martin’s Axiom or PFA? A negative answer to this problem would follow from the positive answer to: Problem Is σG0 = σD0? Remark For any tame Gδ-set G ⊂ I its countable power G ω is not tame in Iω. Is G ω ∈ σG0?

T.Banakh Topologically invariant σ-ideals on Polish spaces

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 under Martin’s Axiom or PFA? A negative answer to this problem would follow from the positive answer to: Problem Is σG0 = σD0? Remark For any tame Gδ-set G ⊂ I its countable power G ω is not tame in Iω. Is G ω ∈ σG0?

T.Banakh Topologically invariant σ-ideals on Polish spaces

Another Open Problem

Theorem (Classics) Each linear Borel subspace L in ℓ2 is meager and hence belongs to the ideal M = σZ0. Theorem (Banakh, 1999; Dobrowolski, Marciszewski, 2002) There is a linear Borel subspace L in ℓ2 which is does not belong to the σ-ideal σZω. Problem Let L be a linear Borel subspace in ℓ2. Is L ∈ σZn for all n ∈ ω? Is L ∈ σZ1?

T.Banakh Topologically invariant σ-ideals on Polish spaces

Another Open Problem

Theorem (Classics) Each linear Borel subspace L in ℓ2 is meager and hence belongs to the ideal M = σZ0. Theorem (Banakh, 1999; Dobrowolski, Marciszewski, 2002) There is a linear Borel subspace L in ℓ2 which is does not belong to the σ-ideal σZω. Problem Let L be a linear Borel subspace in ℓ2. Is L ∈ σZn for all n ∈ ω? Is L ∈ σZ1?

T.Banakh Topologically invariant σ-ideals on Polish spaces

Another Open Problem

Theorem (Classics) Each linear Borel subspace L in ℓ2 is meager and hence belongs to the ideal M = σZ0. Theorem (Banakh, 1999; Dobrowolski, Marciszewski, 2002) There is a linear Borel subspace L in ℓ2 which is does not belong to the σ-ideal σZω. Problem Let L be a linear Borel subspace in ℓ2. Is L ∈ σZn for all n ∈ ω? Is L ∈ σZ1?

T.Banakh Topologically invariant σ-ideals on Polish spaces

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.Repovˇ s, Universal nowhere dense subsets in locally compact manifolds, preprint (http://arxiv.org/abs/1302.5651). T.Banakh, D.Repovˇ s, Universal meager Fσ-sets in locally compact manifolds, to appear in CMUC (http://arxiv.org/abs/1302.5653).

T.Banakh Topologically invariant σ-ideals on Polish spaces

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.Repovˇ s, Universal nowhere dense subsets in locally compact manifolds, preprint (http://arxiv.org/abs/1302.5651). T.Banakh, D.Repovˇ s, Universal meager Fσ-sets in locally compact manifolds, to appear in CMUC (http://arxiv.org/abs/1302.5653).

T.Banakh Topologically invariant σ-ideals on Polish spaces

Thanks!

¨ ⌣

T.Banakh Topologically invariant σ-ideals on Polish spaces