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

1-generated topologically invariant ideals Each family F of subsets of a topological space X generates the topologically invariant ideal A ⊂ X : ∃ ( h n ) n ∈ ω ∈ H ( X ) ω , ( F n ) n ∈ ω ∈ F ω A ⊂ � � � σ F = h n ( F n ) . n ∈ ω A topologically invariant σ -ideal I on X is called 1-generated if I = σ F 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 M 1 0 ⊂ R . [Menger, 1926]: The ideal M of meager subset of R n is 1-generated by the Menger cube M n n − 1 ⊂ R n . s, 2012]: The ideal M of meager subset of [Banakh, Repovˇ the Hilbert cube I ω 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 A ⊂ X : ∃ ( h n ) n ∈ ω ∈ H ( X ) ω , ( F n ) n ∈ ω ∈ F ω A ⊂ � � � σ F = h n ( F n ) . n ∈ ω A topologically invariant σ -ideal I on X is called 1-generated if I = σ F 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 M 1 0 ⊂ R . [Menger, 1926]: The ideal M of meager subset of R n is 1-generated by the Menger cube M n n − 1 ⊂ R n . s, 2012]: The ideal M of meager subset of [Banakh, Repovˇ the Hilbert cube I ω 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 A ⊂ X : ∃ ( h n ) n ∈ ω ∈ H ( X ) ω , ( F n ) n ∈ ω ∈ F ω A ⊂ � � � σ F = h n ( F n ) . n ∈ ω A topologically invariant σ -ideal I on X is called 1-generated if I = σ F 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 M 1 0 ⊂ R . [Menger, 1926]: The ideal M of meager subset of R n is 1-generated by the Menger cube M n n − 1 ⊂ R n . s, 2012]: The ideal M of meager subset of [Banakh, Repovˇ the Hilbert cube I ω 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 A ⊂ X : ∃ ( h n ) n ∈ ω ∈ H ( X ) ω , ( F n ) n ∈ ω ∈ F ω A ⊂ � � � σ F = h n ( F n ) . n ∈ ω A topologically invariant σ -ideal I on X is called 1-generated if I = σ F 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 M 1 0 ⊂ R . [Menger, 1926]: The ideal M of meager subset of R n is 1-generated by the Menger cube M n n − 1 ⊂ R n . s, 2012]: The ideal M of meager subset of [Banakh, Repovˇ the Hilbert cube I ω is 1-generated. T.Banakh Topologically invariant σ -ideals on Polish spaces

Different faces of the ideal M : the ideals σ Z n A closed subset A of a topological space X is called a Z n -set for n ≤ ω if the set { f ∈ C ( I n , X ) : f ( I n ) ∩ A = ∅} is dense in the function space C ( I n , X ) endowed with the compact-open topology. Fact A subset A ⊂ X is a Z 0 -set iff A is closed and nowhere dense in X. The family Z n of all Z n -sets in X generates the topologically invariant σ -ideal σ Z n having F σ -base. Fact M = σ Z 0 ⊃ σ Z 1 ⊃ σ Z 2 ⊃ · · · ⊃ σ Z ω . T.Banakh Topologically invariant σ -ideals on Polish spaces

Different faces of the ideal M : the ideals σ Z n A closed subset A of a topological space X is called a Z n -set for n ≤ ω if the set { f ∈ C ( I n , X ) : f ( I n ) ∩ A = ∅} is dense in the function space C ( I n , X ) endowed with the compact-open topology. Fact A subset A ⊂ X is a Z 0 -set iff A is closed and nowhere dense in X. The family Z n of all Z n -sets in X generates the topologically invariant σ -ideal σ Z n having F σ -base. Fact M = σ Z 0 ⊃ σ Z 1 ⊃ σ Z 2 ⊃ · · · ⊃ σ Z ω . T.Banakh Topologically invariant σ -ideals on Polish spaces

Different faces of the ideal M : the ideals σ Z n A closed subset A of a topological space X is called a Z n -set for n ≤ ω if the set { f ∈ C ( I n , X ) : f ( I n ) ∩ A = ∅} is dense in the function space C ( I n , X ) endowed with the compact-open topology. Fact A subset A ⊂ X is a Z 0 -set iff A is closed and nowhere dense in X. The family Z n of all Z n -sets in X generates the topologically invariant σ -ideal σ Z n having F σ -base. Fact M = σ Z 0 ⊃ σ Z 1 ⊃ σ Z 2 ⊃ · · · ⊃ σ Z ω . T.Banakh Topologically invariant σ -ideals on Polish spaces

Different faces of the ideal M : the ideals σ Z n A closed subset A of a topological space X is called a Z n -set for n ≤ ω if the set { f ∈ C ( I n , X ) : f ( I n ) ∩ A = ∅} is dense in the function space C ( I n , X ) endowed with the compact-open topology. Fact A subset A ⊂ X is a Z 0 -set iff A is closed and nowhere dense in X. The family Z n of all Z n -sets in X generates the topologically invariant σ -ideal σ Z n having F σ -base. Fact M = σ Z 0 ⊃ σ Z 1 ⊃ σ Z 2 ⊃ · · · ⊃ σ Z ω . T.Banakh Topologically invariant σ -ideals on Polish spaces

The ideals σ D n and σ D n For a topological space X and an integer number n ∈ ω let D n (resp. D n ) be the family of all at most n -dimensional (closed) subsets of X ; D <ω = � n ∈ ω D n , D <ω = � n ∈ ω D n ; σ D n , σ D <ω , σ D n , σ D <ω be the σ -ideals generated by the families D n , D <ω , D n , D <ω . Fact For a metrizable separable space X the σ -ideal: σ D 0 = σ D <ω contains all countably-dimensional subsets of X; σ D <ω contains all strongly countably-dimensional sets in X; σ D 0 ⊂ σ D 1 ⊂ σ D 2 ⊂ · · · ⊂ σ D <ω ⊂ σ D <ω = σ D 0 . For X = R n , we get M = σ D n − 1 = σ Z 0 . T.Banakh Topologically invariant σ -ideals on Polish spaces

The ideals σ D n and σ D n For a topological space X and an integer number n ∈ ω let D n (resp. D n ) be the family of all at most n -dimensional (closed) subsets of X ; D <ω = � n ∈ ω D n , D <ω = � n ∈ ω D n ; σ D n , σ D <ω , σ D n , σ D <ω be the σ -ideals generated by the families D n , D <ω , D n , D <ω . Fact For a metrizable separable space X the σ -ideal: σ D 0 = σ D <ω contains all countably-dimensional subsets of X; σ D <ω contains all strongly countably-dimensional sets in X; σ D 0 ⊂ σ D 1 ⊂ σ D 2 ⊂ · · · ⊂ σ D <ω ⊂ σ D <ω = σ D 0 . For X = R n , we get M = σ D n − 1 = σ Z 0 . T.Banakh Topologically invariant σ -ideals on Polish spaces

The ideals σ D n and σ D n For a topological space X and an integer number n ∈ ω let D n (resp. D n ) be the family of all at most n -dimensional (closed) subsets of X ; D <ω = � n ∈ ω D n , D <ω = � n ∈ ω D n ; σ D n , σ D <ω , σ D n , σ D <ω be the σ -ideals generated by the families D n , D <ω , D n , D <ω . Fact For a metrizable separable space X the σ -ideal: σ D 0 = σ D <ω contains all countably-dimensional subsets of X; σ D <ω contains all strongly countably-dimensional sets in X; σ D 0 ⊂ σ D 1 ⊂ σ D 2 ⊂ · · · ⊂ σ D <ω ⊂ σ D <ω = σ D 0 . For X = R n , we get M = σ D n − 1 = σ Z 0 . T.Banakh Topologically invariant σ -ideals on Polish spaces

The ideals σ D n and σ D n For a topological space X and an integer number n ∈ ω let D n (resp. D n ) be the family of all at most n -dimensional (closed) subsets of X ; D <ω = � n ∈ ω D n , D <ω = � n ∈ ω D n ; σ D n , σ D <ω , σ D n , σ D <ω be the σ -ideals generated by the families D n , D <ω , D n , D <ω . Fact For a metrizable separable space X the σ -ideal: σ D 0 = σ D <ω contains all countably-dimensional subsets of X; σ D <ω contains all strongly countably-dimensional sets in X; σ D 0 ⊂ σ D 1 ⊂ σ D 2 ⊂ · · · ⊂ σ D <ω ⊂ σ D <ω = σ D 0 . For X = R n , we get M = σ D n − 1 = σ Z 0 . T.Banakh Topologically invariant σ -ideals on Polish spaces

The ideals σ D n and σ D n For a topological space X and an integer number n ∈ ω let D n (resp. D n ) be the family of all at most n -dimensional (closed) subsets of X ; D <ω = � n ∈ ω D n , D <ω = � n ∈ ω D n ; σ D n , σ D <ω , σ D n , σ D <ω be the σ -ideals generated by the families D n , D <ω , D n , D <ω . Fact For a metrizable separable space X the σ -ideal: σ D 0 = σ D <ω contains all countably-dimensional subsets of X; σ D <ω contains all strongly countably-dimensional sets in X; σ D 0 ⊂ σ D 1 ⊂ σ D 2 ⊂ · · · ⊂ σ D <ω ⊂ σ D <ω = σ D 0 . For X = R n , we get M = σ D n − 1 = σ Z 0 . T.Banakh Topologically invariant σ -ideals on Polish spaces

The ideals σ D n and σ D n For a topological space X and an integer number n ∈ ω let D n (resp. D n ) be the family of all at most n -dimensional (closed) subsets of X ; D <ω = � n ∈ ω D n , D <ω = � n ∈ ω D n ; σ D n , σ D <ω , σ D n , σ D <ω be the σ -ideals generated by the families D n , D <ω , D n , D <ω . Fact For a metrizable separable space X the σ -ideal: σ D 0 = σ D <ω contains all countably-dimensional subsets of X; σ D <ω contains all strongly countably-dimensional sets in X; σ D 0 ⊂ σ D 1 ⊂ σ D 2 ⊂ · · · ⊂ σ D <ω ⊂ σ D <ω = σ D 0 . For X = R n , we get M = σ D n − 1 = σ Z 0 . T.Banakh Topologically invariant σ -ideals on Polish spaces

The ideals σ D n and σ D n For a topological space X and an integer number n ∈ ω let D n (resp. D n ) be the family of all at most n -dimensional (closed) subsets of X ; D <ω = � n ∈ ω D n , D <ω = � n ∈ ω D n ; σ D n , σ D <ω , σ D n , σ D <ω be the σ -ideals generated by the families D n , D <ω , D n , D <ω . Fact For a metrizable separable space X the σ -ideal: σ D 0 = σ D <ω contains all countably-dimensional subsets of X; σ D <ω contains all strongly countably-dimensional sets in X; σ D 0 ⊂ σ D 1 ⊂ σ D 2 ⊂ · · · ⊂ σ D <ω ⊂ σ D <ω = σ D 0 . For X = R n , we get M = σ D n − 1 = σ Z 0 . T.Banakh Topologically invariant σ -ideals on Polish spaces

The ideals σ D n and σ D n For a topological space X and an integer number n ∈ ω let D n (resp. D n ) be the family of all at most n -dimensional (closed) subsets of X ; D <ω = � n ∈ ω D n , D <ω = � n ∈ ω D n ; σ D n , σ D <ω , σ D n , σ D <ω be the σ -ideals generated by the families D n , D <ω , D n , D <ω . Fact For a metrizable separable space X the σ -ideal: σ D 0 = σ D <ω contains all countably-dimensional subsets of X; σ D <ω contains all strongly countably-dimensional sets in X; σ D 0 ⊂ σ D 1 ⊂ σ D 2 ⊂ · · · ⊂ σ D <ω ⊂ σ D <ω = σ D 0 . For X = R n , we get M = σ D n − 1 = σ Z 0 . 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 σ D n , σ Z m and their intersections σ D n ∩ σ Z m . 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 σ D n , σ Z m and their intersections σ D n ∩ σ Z m . 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 R n , I ω , 2 ω , ω ω are topologically homogeneous. Theorem (folklore) Each infinite zero-dimensional topologically homogeneous Polish space X is homeomorphic to: ω iff X is countable; 2 ω iff X is uncountable and compact; 2 ω × ω iff X is uncountable, locally compact, and not compact; ω ω 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 R n , I ω , 2 ω , ω ω are topologically homogeneous. Theorem (folklore) Each infinite zero-dimensional topologically homogeneous Polish space X is homeomorphic to: ω iff X is countable; 2 ω iff X is uncountable and compact; 2 ω × ω iff X is uncountable, locally compact, and not compact; ω ω 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 R n , I ω , 2 ω , ω ω are topologically homogeneous. Theorem (folklore) Each infinite zero-dimensional topologically homogeneous Polish space X is homeomorphic to: ω iff X is countable; 2 ω iff X is uncountable and compact; 2 ω × ω iff X is uncountable, locally compact, and not compact; ω ω 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 on a zero-dimensional topologically homogeneous Polish space X is equal to M or σ K . The σ -ideals M and σ K 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 on a zero-dimensional topologically homogeneous Polish space X is equal to M or σ K . The σ -ideals M and σ K 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 F σ -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 F σ -base on X. T.Banakh Topologically invariant σ -ideals on Polish spaces

The ideal M on Euclidean spaces lowski, ˙ Theorem (Banakh, Morayne, Ra� Zeberski, 2011) Each non-trivial topologically invariant σ -ideal with BP-base on R n is contained in the ideal M . So, M is the largest non-trivial topologically invariant σ -ideal with BP-base on R n . For the Hilbert cube I ω this is not true anymore. Example The σ -ideal σ D 0 of countably-dimensional subset in the Hilbert cube I ω is non-trivial, topologically invariant, has G δσ -base, but σ D 0 �⊂ M . T.Banakh Topologically invariant σ -ideals on Polish spaces

The ideal M on Euclidean spaces lowski, ˙ Theorem (Banakh, Morayne, Ra� Zeberski, 2011) Each non-trivial topologically invariant σ -ideal with BP-base on R n is contained in the ideal M . So, M is the largest non-trivial topologically invariant σ -ideal with BP-base on R n . For the Hilbert cube I ω this is not true anymore. Example The σ -ideal σ D 0 of countably-dimensional subset in the Hilbert cube I ω is non-trivial, topologically invariant, has G δσ -base, but σ D 0 �⊂ M . T.Banakh Topologically invariant σ -ideals on Polish spaces

The ideal M on Euclidean spaces lowski, ˙ Theorem (Banakh, Morayne, Ra� Zeberski, 2011) Each non-trivial topologically invariant σ -ideal with BP-base on R n is contained in the ideal M . So, M is the largest non-trivial topologically invariant σ -ideal with BP-base on R n . For the Hilbert cube I ω this is not true anymore. Example The σ -ideal σ D 0 of countably-dimensional subset in the Hilbert cube I ω is non-trivial, topologically invariant, has G δσ -base, but σ D 0 �⊂ M . T.Banakh Topologically invariant σ -ideals on Polish spaces

The ideal M on Euclidean spaces lowski, ˙ Theorem (Banakh, Morayne, Ra� Zeberski, 2011) Each non-trivial topologically invariant σ -ideal with BP-base on R n is contained in the ideal M . So, M is the largest non-trivial topologically invariant σ -ideal with BP-base on R n . For the Hilbert cube I ω this is not true anymore. Example The σ -ideal σ D 0 of countably-dimensional subset in the Hilbert cube I ω is non-trivial, topologically invariant, has G δσ -base, but σ D 0 �⊂ M . T.Banakh Topologically invariant σ -ideals on Polish spaces

The σ -ideal σ C 0 generated by minimal Cantor sets A Cantor set in a topological space X is any subset C ⊂ X homeomorphic to the Cantor cube 2 ω . 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 σ C 0 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 on a Polish space X contains the σ -ideal σ C 0 generated by minimal Cantor sets in X . T.Banakh Topologically invariant σ -ideals on Polish spaces

The σ -ideal σ C 0 generated by minimal Cantor sets A Cantor set in a topological space X is any subset C ⊂ X homeomorphic to the Cantor cube 2 ω . 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 σ C 0 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 on a Polish space X contains the σ -ideal σ C 0 generated by minimal Cantor sets in X . T.Banakh Topologically invariant σ -ideals on Polish spaces

The σ -ideal σ C 0 generated by minimal Cantor sets A Cantor set in a topological space X is any subset C ⊂ X homeomorphic to the Cantor cube 2 ω . 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 σ C 0 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 on a Polish space X contains the σ -ideal σ C 0 generated by minimal Cantor sets in X . T.Banakh Topologically invariant σ -ideals on Polish spaces

The σ -ideal σ C 0 generated by minimal Cantor sets A Cantor set in a topological space X is any subset C ⊂ X homeomorphic to the Cantor cube 2 ω . 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 σ C 0 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 on a Polish space X contains the σ -ideal σ C 0 generated by minimal Cantor sets in X . T.Banakh Topologically invariant σ -ideals on Polish spaces

The σ -ideal σ G 0 generated by minimal dense G δ -sets A dense G δ -set A in a Polish space X is called a minimal dense G δ -set in X if for each dense G δ -set B ⊂ X there is a homeomorphism h : X → X such that h ( A ) ⊂ B . The σ -ideal σ G 0 generated by minimal dense G δ -sets in X is 1-generated (by any minimal dense G δ -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 σ G 0 generated by minimal dense G δ -sets in X . T.Banakh Topologically invariant σ -ideals on Polish spaces

The σ -ideal σ G 0 generated by minimal dense G δ -sets A dense G δ -set A in a Polish space X is called a minimal dense G δ -set in X if for each dense G δ -set B ⊂ X there is a homeomorphism h : X → X such that h ( A ) ⊂ B . The σ -ideal σ G 0 generated by minimal dense G δ -sets in X is 1-generated (by any minimal dense G δ -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 σ G 0 generated by minimal dense G δ -sets in X . T.Banakh Topologically invariant σ -ideals on Polish spaces

The σ -ideal σ G 0 generated by minimal dense G δ -sets A dense G δ -set A in a Polish space X is called a minimal dense G δ -set in X if for each dense G δ -set B ⊂ X there is a homeomorphism h : X → X such that h ( A ) ⊂ B . The σ -ideal σ G 0 generated by minimal dense G δ -sets in X is 1-generated (by any minimal dense G δ -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 σ G 0 generated by minimal dense G δ -sets in X . T.Banakh Topologically invariant σ -ideals on Polish spaces

Minimality of the σ -ideals σ C 0 and σ G 0 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 σ C 0 ⊂ I ; 2 If I �⊂ M , then σ G 0 ⊂ I . Problem Given a Polish space X, study the σ -ideals σ C 0 and σ G 0 generated by minimal Cantor sets and minimal dense G δ -sets, respectively. T.Banakh Topologically invariant σ -ideals on Polish spaces

Minimality of the σ -ideals σ C 0 and σ G 0 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 σ C 0 ⊂ I ; 2 If I �⊂ M , then σ G 0 ⊂ I . Problem Given a Polish space X, study the σ -ideals σ C 0 and σ G 0 generated by minimal Cantor sets and minimal dense G δ -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 R n is minimal if and only if C is tame, which means that h ( C ) ⊂ R × { 0 } n − 1 for some homeomorphism h of R n . Thus for a Euclidean space X = R n the ideal σ C 0 is not trivial. It is known that each Cantor set in R n for n ≤ 2 is tame. A Cantor set C ⊂ R n 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 R n is minimal if and only if C is tame, which means that h ( C ) ⊂ R × { 0 } n − 1 for some homeomorphism h of R n . Thus for a Euclidean space X = R n the ideal σ C 0 is not trivial. It is known that each Cantor set in R n for n ≤ 2 is tame. A Cantor set C ⊂ R n 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 R n is minimal if and only if C is tame, which means that h ( C ) ⊂ R × { 0 } n − 1 for some homeomorphism h of R n . Thus for a Euclidean space X = R n the ideal σ C 0 is not trivial. It is known that each Cantor set in R n for n ≤ 2 is tame. A Cantor set C ⊂ R n 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 R n is minimal if and only if C is tame, which means that h ( C ) ⊂ R × { 0 } n − 1 for some homeomorphism h of R n . Thus for a Euclidean space X = R n the ideal σ C 0 is not trivial. It is known that each Cantor set in R n for n ≤ 2 is tame. A Cantor set C ⊂ R n 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 R n is minimal if and only if C is tame, which means that h ( C ) ⊂ R × { 0 } n − 1 for some homeomorphism h of R n . Thus for a Euclidean space X = R n the ideal σ C 0 is not trivial. It is known that each Cantor set in R n for n ≤ 2 is tame. A Cantor set C ⊂ R n 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 R n is minimal if and only if C is tame, which means that h ( C ) ⊂ R × { 0 } n − 1 for some homeomorphism h of R n . Thus for a Euclidean space X = R n the ideal σ C 0 is not trivial. It is known that each Cantor set in R n for n ≤ 2 is tame. A Cantor set C ⊂ R n which is not tame is called wild. T.Banakh Topologically invariant σ -ideals on Polish spaces

A wild Cantor set in R 3 : the Antoine’s necklace T.Banakh Topologically invariant σ -ideals on Polish spaces

Z -set characterization of minimal Cantor sets in R n . Theorem (McMillan, 1964) For a Cantor set C ⊂ R n the following are equivalent: 1 C is a minimal Cantor set in R n ; 2 C is tame in R n ; 3 C is a Z k -set in R n for all k < n; 4 C is a Z k -set in R n for k = min { 2 , n − 1 } . Corollary If X = R n , then σ C 0 = σ D 0 ∩ σ Z k for k = min { 2 , n − 1 } . T.Banakh Topologically invariant σ -ideals on Polish spaces

Z -set characterization of minimal Cantor sets in R n . Theorem (McMillan, 1964) For a Cantor set C ⊂ R n the following are equivalent: 1 C is a minimal Cantor set in R n ; 2 C is tame in R n ; 3 C is a Z k -set in R n for all k < n; 4 C is a Z k -set in R n for k = min { 2 , n − 1 } . Corollary If X = R n , then σ C 0 = σ D 0 ∩ σ Z k for k = min { 2 , n − 1 } . T.Banakh Topologically invariant σ -ideals on Polish spaces

Z -sets in the Hilbert cube I ω . Theorem (Unknotting Z -sets) Any homeomorphism h : A → B between Z ω -sets in I ω can be h : I ω → I ω of I ω . extended to a homeomorphism ¯ Theorem (Kroonenberg, 1974) A finite-dimensional subset A ⊂ I ω is a Z ω -set iff it is a Z 2 -set. These two theorems imply: T.Banakh Topologically invariant σ -ideals on Polish spaces

Z -sets in the Hilbert cube I ω . Theorem (Unknotting Z -sets) Any homeomorphism h : A → B between Z ω -sets in I ω can be h : I ω → I ω of I ω . extended to a homeomorphism ¯ Theorem (Kroonenberg, 1974) A finite-dimensional subset A ⊂ I ω is a Z ω -set iff it is a Z 2 -set. These two theorems imply: T.Banakh Topologically invariant σ -ideals on Polish spaces

Z -sets in the Hilbert cube I ω . Theorem (Unknotting Z -sets) Any homeomorphism h : A → B between Z ω -sets in I ω can be h : I ω → I ω of I ω . extended to a homeomorphism ¯ Theorem (Kroonenberg, 1974) A finite-dimensional subset A ⊂ I ω is a Z ω -set iff it is a Z 2 -set. These two theorems imply: T.Banakh Topologically invariant σ -ideals on Polish spaces

Z -sets in the Hilbert cube I ω . Theorem (Unknotting Z -sets) Any homeomorphism h : A → B between Z ω -sets in I ω can be h : I ω → I ω of I ω . extended to a homeomorphism ¯ Theorem (Kroonenberg, 1974) A finite-dimensional subset A ⊂ I ω is a Z ω -set iff it is a Z 2 -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 ⊂ I ω the following are equivalent: 1 C is a minimal Cantor set in I ω ; 2 C is a Z ω -set in I ω ; 3 C is a Z 2 -set in I ω . Corollary σ C 0 = σ D 0 ∩ σ Z ω = σ D 0 ∩ σ Z ω is the smallest non-trivial topologically invariant σ -ideal with analytic base on the Hilbert cube I ω . T.Banakh Topologically invariant σ -ideals on Polish spaces

Characterizing minimal Cantor sets in the Hilbert cube Theorem For a Cantor set C ⊂ I ω the following are equivalent: 1 C is a minimal Cantor set in I ω ; 2 C is a Z ω -set in I ω ; 3 C is a Z 2 -set in I ω . Corollary σ C 0 = σ D 0 ∩ σ Z ω = σ D 0 ∩ σ Z ω is the smallest non-trivial topologically invariant σ -ideal with analytic base on the Hilbert cube I ω . T.Banakh Topologically invariant σ -ideals on Polish spaces

Now we shall consider minimal dense G δ -sets in I m -manifolds. We start with minimal (dense) open sets in I m -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 G δ -sets in I m -manifolds. We start with minimal (dense) open sets in I m -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 G δ -sets in I m -manifolds. We start with minimal (dense) open sets in I m -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

I m -manifolds Let E be a topological space. An E -manifold is a paracompact topological space that has a cover by open sets homeomorphic to open subspaces of the model space E . So, for m < ω , I m -manifolds are usual m -manifolds with boundary, I ω -manifolds are Hilbert cube manifolds. T.Banakh Topologically invariant σ -ideals on Polish spaces

I m -manifolds Let E be a topological space. An E -manifold is a paracompact topological space that has a cover by open sets homeomorphic to open subspaces of the model space E . So, for m < ω , I m -manifolds are usual m -manifolds with boundary, I ω -manifolds are Hilbert cube manifolds. T.Banakh Topologically invariant σ -ideals on Polish spaces

I m -manifolds Let E be a topological space. An E -manifold is a paracompact topological space that has a cover by open sets homeomorphic to open subspaces of the model space E . So, for m < ω , I m -manifolds are usual m -manifolds with boundary, I ω -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 = { F n } n ∈ ω of subsets of a compact metric space X is vanishing if and only if diam ( F n ) → 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 = { F n } n ∈ ω of subsets of a compact metric space X is vanishing if and only if diam ( F n ) → 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 = { F n } n ∈ ω of subsets of a compact metric space X is vanishing if and only if diam ( F n ) → 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 I m -manifolds Let X be an I m -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 � ( R m , I m ) if m < ω ( I ω × [0 , ∞ ) , I ω × [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 I m -manifold X is a minimal open set in X. T.Banakh Topologically invariant σ -ideals on Polish spaces

Tame balls in I m -manifolds Let X be an I m -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 � ( R m , I m ) if m < ω ( I ω × [0 , ∞ ) , I ω × [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 I m -manifold X is a minimal open set in X. T.Banakh Topologically invariant σ -ideals on Polish spaces

Tame balls in I m -manifolds Let X be an I m -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 � ( R m , I m ) if m < ω ( I ω × [0 , ∞ ) , I ω × [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 I m -manifold X is a minimal open set in X. T.Banakh Topologically invariant σ -ideals on Polish spaces

Tame balls in I m -manifolds Let X be an I m -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 � ( R m , I m ) if m < ω ( I ω × [0 , ∞ ) , I ω × [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 I m -manifold X is a minimal open set in X. T.Banakh Topologically invariant σ -ideals on Polish spaces

Minimal and tame open sets in I m -manifolds An open set U ⊂ X is called tame open set in an I m -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 I m -manifold X contains a dense tame open set. 2 Any two dense tame open sets U , V in an I m -manifold X are ambiently homeomorphic in X. Corollary Each dense tame open set in an I m -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 I m -manifolds An open set U ⊂ X is called tame open set in an I m -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 I m -manifold X contains a dense tame open set. 2 Any two dense tame open sets U , V in an I m -manifold X are ambiently homeomorphic in X. Corollary Each dense tame open set in an I m -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 I m -manifolds An open set U ⊂ X is called tame open set in an I m -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 I m -manifold X contains a dense tame open set. 2 Any two dense tame open sets U , V in an I m -manifold X are ambiently homeomorphic in X. Corollary Each dense tame open set in an I m -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 I m -manifolds An open set U ⊂ X is called tame open set in an I m -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 I m -manifold X contains a dense tame open set. 2 Any two dense tame open sets U , V in an I m -manifold X are ambiently homeomorphic in X. Corollary Each dense tame open set in an I m -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 G δ -sets in I m -manifolds A subset G of an I m -manifold X is called a tame G δ -set in X if G = � ∞ U for some vanishing ultrafamily U of tame open balls in � ∞ U = � { � ( U \ F ) : F ⊂ U , |F| < ∞} . X . Here It follows that each tame G δ -set G = � n ∈ ω U n for some decreasing family of tame open sets U n . Theorem (Banakh-Repovˇ s, 2012) 1 Any dense G δ -subset of an I m -manifold X contains a dense tame G δ -set in X. 2 Any two dense tame G δ -sets in an I m -manifold X are ambiently homeomorphic. Corollary Each dense tame G δ -set in an I m -manifold X is a minimal dense G δ -set in X. T.Banakh Topologically invariant σ -ideals on Polish spaces

Minimal and tame G δ -sets in I m -manifolds A subset G of an I m -manifold X is called a tame G δ -set in X if G = � ∞ U for some vanishing ultrafamily U of tame open balls in � ∞ U = � { � ( U \ F ) : F ⊂ U , |F| < ∞} . X . Here It follows that each tame G δ -set G = � n ∈ ω U n for some decreasing family of tame open sets U n . Theorem (Banakh-Repovˇ s, 2012) 1 Any dense G δ -subset of an I m -manifold X contains a dense tame G δ -set in X. 2 Any two dense tame G δ -sets in an I m -manifold X are ambiently homeomorphic. Corollary Each dense tame G δ -set in an I m -manifold X is a minimal dense G δ -set in X. T.Banakh Topologically invariant σ -ideals on Polish spaces

Minimal and tame G δ -sets in I m -manifolds A subset G of an I m -manifold X is called a tame G δ -set in X if G = � ∞ U for some vanishing ultrafamily U of tame open balls in � ∞ U = � { � ( U \ F ) : F ⊂ U , |F| < ∞} . X . Here It follows that each tame G δ -set G = � n ∈ ω U n for some decreasing family of tame open sets U n . Theorem (Banakh-Repovˇ s, 2012) 1 Any dense G δ -subset of an I m -manifold X contains a dense tame G δ -set in X. 2 Any two dense tame G δ -sets in an I m -manifold X are ambiently homeomorphic. Corollary Each dense tame G δ -set in an I m -manifold X is a minimal dense G δ -set in X. T.Banakh Topologically invariant σ -ideals on Polish spaces

Minimal and tame G δ -sets in I m -manifolds A subset G of an I m -manifold X is called a tame G δ -set in X if G = � ∞ U for some vanishing ultrafamily U of tame open balls in � ∞ U = � { � ( U \ F ) : F ⊂ U , |F| < ∞} . X . Here It follows that each tame G δ -set G = � n ∈ ω U n for some decreasing family of tame open sets U n . Theorem (Banakh-Repovˇ s, 2012) 1 Any dense G δ -subset of an I m -manifold X contains a dense tame G δ -set in X. 2 Any two dense tame G δ -sets in an I m -manifold X are ambiently homeomorphic. Corollary Each dense tame G δ -set in an I m -manifold X is a minimal dense G δ -set in X. T.Banakh Topologically invariant σ -ideals on Polish spaces

Minimal and tame G δ -sets in I m -manifolds A subset G of an I m -manifold X is called a tame G δ -set in X if G = � ∞ U for some vanishing ultrafamily U of tame open balls in � ∞ U = � { � ( U \ F ) : F ⊂ U , |F| < ∞} . X . Here It follows that each tame G δ -set G = � n ∈ ω U n for some decreasing family of tame open sets U n . Theorem (Banakh-Repovˇ s, 2012) 1 Any dense G δ -subset of an I m -manifold X contains a dense tame G δ -set in X. 2 Any two dense tame G δ -sets in an I m -manifold X are ambiently homeomorphic. Corollary Each dense tame G δ -set in an I m -manifold X is a minimal dense G δ -set in X. T.Banakh Topologically invariant σ -ideals on Polish spaces

Characterizing minimal dense G δ -sets in I m -manifolds Theorem (Banakh-Repovˇ s, 2012) A dense G δ -set G in an I m -manifold X is minimal if and only if G is tame in X. Corollary The ideal M of meager subsets in any I m -manifold X is 1-generated (by the complement of any dense tame G δ -set in X). T.Banakh Topologically invariant σ -ideals on Polish spaces

Characterizing minimal dense G δ -sets in I m -manifolds Theorem (Banakh-Repovˇ s, 2012) A dense G δ -set G in an I m -manifold X is minimal if and only if G is tame in X. Corollary The ideal M of meager subsets in any I m -manifold X is 1-generated (by the complement of any dense tame G δ -set in X). T.Banakh Topologically invariant σ -ideals on Polish spaces