Pinning Down versus Density Lajos Soukup Alfrd Rnyi Institute of - PowerPoint PPT Presentation

• X = � ω ω × ω, τ � • X n = ( ω n \ ω n − 1 ) × ω . • P = � ( ω n \ ω n − 1 ) . � � �� � If n ∈ ω , f ∈ P , A ⊂ ω let G ( n , f , A )= ω m \ f ( m ) × A . m ≥ n � � ω . Fix an independent family A = { A n , f : n ∈ ω, f ∈ P }⊂ ω Clopen subbase: { G ( n , f , A n , f ) : n ∈ ω, f ∈ P . } If ∅ � = U ⊂ open X then G ( n , f , A ) ⊂ U for some n ∈ ω , f ∈ � , A U ∈ �A�

• X = � ω ω × ω, τ � • X n = ( ω n \ ω n − 1 ) × ω . • P = � ( ω n \ ω n − 1 ) . � � �� � If n ∈ ω , f ∈ P , A ⊂ ω let G ( n , f , A )= ω m \ f ( m ) × A . m ≥ n � � ω . Fix an independent family A = { A n , f : n ∈ ω, f ∈ P }⊂ ω Clopen subbase: { G ( n , f , A n , f ) : n ∈ ω, f ∈ P . } If ∅ � = U ⊂ open X then G ( n , f , A ) ⊂ U for some n ∈ ω , f ∈ � , A U ∈ �A� Claim: d ( X ) = ω ω .

• X = � ω ω × ω, τ � • X n = ( ω n \ ω n − 1 ) × ω . • P = � ( ω n \ ω n − 1 ) . � � �� � If n ∈ ω , f ∈ P , A ⊂ ω let G ( n , f , A )= ω m \ f ( m ) × A . m ≥ n � � ω . Fix an independent family A = { A n , f : n ∈ ω, f ∈ P }⊂ ω Clopen subbase: { G ( n , f , A n , f ) : n ∈ ω, f ∈ P . } If ∅ � = U ⊂ open X then G ( n , f , A ) ⊂ U for some n ∈ ω , f ∈ � , A U ∈ �A� Claim: d ( X ) = ω ω . • Assume | D | < ω ω .

• X = � ω ω × ω, τ � • X n = ( ω n \ ω n − 1 ) × ω . • P = � ( ω n \ ω n − 1 ) . � � �� � If n ∈ ω , f ∈ P , A ⊂ ω let G ( n , f , A )= ω m \ f ( m ) × A . m ≥ n � � ω . Fix an independent family A = { A n , f : n ∈ ω, f ∈ P }⊂ ω Clopen subbase: { G ( n , f , A n , f ) : n ∈ ω, f ∈ P . } If ∅ � = U ⊂ open X then G ( n , f , A ) ⊂ U for some n ∈ ω , f ∈ � , A U ∈ �A� Claim: d ( X ) = ω ω . • Assume | D | < ω ω . • | D | < ω n for some n

• X = � ω ω × ω, τ � • X n = ( ω n \ ω n − 1 ) × ω . • P = � ( ω n \ ω n − 1 ) . � � �� � If n ∈ ω , f ∈ P , A ⊂ ω let G ( n , f , A )= ω m \ f ( m ) × A . m ≥ n � � ω . Fix an independent family A = { A n , f : n ∈ ω, f ∈ P }⊂ ω Clopen subbase: { G ( n , f , A n , f ) : n ∈ ω, f ∈ P . } If ∅ � = U ⊂ open X then G ( n , f , A ) ⊂ U for some n ∈ ω , f ∈ � , A U ∈ �A� Claim: d ( X ) = ω ω . • Assume | D | < ω ω . • | D | < ω n for some n • there is f ∈ P such that D ∩ X m ⊂ f ( m ) × ω for m ≥ n .

• X = � ω ω × ω, τ � • X n = ( ω n \ ω n − 1 ) × ω . • P = � ( ω n \ ω n − 1 ) . � � �� � If n ∈ ω , f ∈ P , A ⊂ ω let G ( n , f , A )= ω m \ f ( m ) × A . m ≥ n � � ω . Fix an independent family A = { A n , f : n ∈ ω, f ∈ P }⊂ ω Clopen subbase: { G ( n , f , A n , f ) : n ∈ ω, f ∈ P . } If ∅ � = U ⊂ open X then G ( n , f , A ) ⊂ U for some n ∈ ω , f ∈ � , A U ∈ �A� Claim: d ( X ) = ω ω . • Assume | D | < ω ω . • | D | < ω n for some n • there is f ∈ P such that D ∩ X m ⊂ f ( m ) × ω for m ≥ n . • Then G ( n , f , A n , f ) ∩ D = ∅ .

• X = � ω ω × ω, τ � • X n = ( ω n \ ω n − 1 ) × ω . • P = � ( ω n \ ω n − 1 ) . � � �� � If n ∈ ω , f ∈ P , A ⊂ ω let G ( n , f , A )= ω m \ f ( m ) × A . m ≥ n � � ω . Fix an independent family A = { A n , f : n ∈ ω, f ∈ P }⊂ ω Clopen subbase: { G ( n , f , A n , f ) : n ∈ ω, f ∈ P . } If ∅ � = U ⊂ open X then G ( n , f , A ) ⊂ U for some n ∈ ω , f ∈ � , A U ∈ �A� Claim: d ( X ) = ω ω . • Assume | D | < ω ω . • | D | < ω n for some n • there is f ∈ P such that D ∩ X m ⊂ f ( m ) × ω for m ≥ n . • Then G ( n , f , A n , f ) ∩ D = ∅ . • Thus D is not dense.

• X = � ω ω × ω, τ � • X n = ( ω n \ ω n − 1 ) × ω . • P = � ( ω n \ ω n − 1 ) . � � �� � If n ∈ ω , f ∈ P , A ⊂ ω let G ( n , f , A )= ω m \ f ( m ) × A . m ≥ n � ω . � Fix an independent family A = { A n , f : n ∈ ω, f ∈ P }⊂ ω Clopen subbase: { G ( n , f , A n , f ) : n ∈ ω, f ∈ P . } If ∅ � = U ⊂ open X then G ( n , f , A ) ⊂ U for some n ∈ ω , f ∈ � , A U ∈ �A� •

• X = � ω ω × ω, τ � • X n = ( ω n \ ω n − 1 ) × ω . • P = � ( ω n \ ω n − 1 ) . � � �� � If n ∈ ω , f ∈ P , A ⊂ ω let G ( n , f , A )= ω m \ f ( m ) × A . m ≥ n � ω . � Fix an independent family A = { A n , f : n ∈ ω, f ∈ P }⊂ ω Clopen subbase: { G ( n , f , A n , f ) : n ∈ ω, f ∈ P . } If ∅ � = U ⊂ open X then G ( n , f , A ) ⊂ U for some n ∈ ω , f ∈ � , A U ∈ �A� Claim: pd ( X ) = ω . •

• X = � ω ω × ω, τ � • X n = ( ω n \ ω n − 1 ) × ω . • P = � ( ω n \ ω n − 1 ) . � � �� � If n ∈ ω , f ∈ P , A ⊂ ω let G ( n , f , A )= ω m \ f ( m ) × A . m ≥ n � ω . � Fix an independent family A = { A n , f : n ∈ ω, f ∈ P }⊂ ω Clopen subbase: { G ( n , f , A n , f ) : n ∈ ω, f ∈ P . } If ∅ � = U ⊂ open X then G ( n , f , A ) ⊂ U for some n ∈ ω , f ∈ � , A U ∈ �A� Claim: pd ( X ) = ω . • U : X → τ be a NEA. •

• X = � ω ω × ω, τ � • X n = ( ω n \ ω n − 1 ) × ω . • P = � ( ω n \ ω n − 1 ) . � � �� � If n ∈ ω , f ∈ P , A ⊂ ω let G ( n , f , A )= ω m \ f ( m ) × A . m ≥ n � ω . � Fix an independent family A = { A n , f : n ∈ ω, f ∈ P }⊂ ω Clopen subbase: { G ( n , f , A n , f ) : n ∈ ω, f ∈ P . } If ∅ � = U ⊂ open X then G ( n , f , A ) ⊂ U for some n ∈ ω , f ∈ � , A U ∈ �A� Claim: pd ( X ) = ω . • U : X → τ be a NEA. • Then U ( p ) ⊃ G ( n p , f p , A p ) for all p ∈ X •

• X = � ω ω × ω, τ � • X n = ( ω n \ ω n − 1 ) × ω . • P = � ( ω n \ ω n − 1 ) . � � �� � If n ∈ ω , f ∈ P , A ⊂ ω let G ( n , f , A )= ω m \ f ( m ) × A . m ≥ n � ω . � Fix an independent family A = { A n , f : n ∈ ω, f ∈ P }⊂ ω Clopen subbase: { G ( n , f , A n , f ) : n ∈ ω, f ∈ P . } If ∅ � = U ⊂ open X then G ( n , f , A ) ⊂ U for some n ∈ ω , f ∈ � , A U ∈ �A� Claim: pd ( X ) = ω . • U : X → τ be a NEA. • Then U ( p ) ⊃ G ( n p , f p , A p ) for all p ∈ X • there is g ∈ P s.t. f p < ∗ g for all p ∈ X . •

• X = � ω ω × ω, τ � • X n = ( ω n \ ω n − 1 ) × ω . • P = � ( ω n \ ω n − 1 ) . � � �� � If n ∈ ω , f ∈ P , A ⊂ ω let G ( n , f , A )= ω m \ f ( m ) × A . m ≥ n � ω . � Fix an independent family A = { A n , f : n ∈ ω, f ∈ P }⊂ ω Clopen subbase: { G ( n , f , A n , f ) : n ∈ ω, f ∈ P . } If ∅ � = U ⊂ open X then G ( n , f , A ) ⊂ U for some n ∈ ω , f ∈ � , A U ∈ �A� Claim: pd ( X ) = ω . • U : X → τ be a NEA. • Then U ( p ) ⊃ G ( n p , f p , A p ) for all p ∈ X • there is g ∈ P s.t. f p < ∗ g for all p ∈ X . • R = { g ( n ) : n ∈ ω } × ω pins down U •

• X = � ω ω × ω, τ � • X n = ( ω n \ ω n − 1 ) × ω . • P = � ( ω n \ ω n − 1 ) . � � �� � If n ∈ ω , f ∈ P , A ⊂ ω let G ( n , f , A )= ω m \ f ( m ) × A . m ≥ n � ω . � Fix an independent family A = { A n , f : n ∈ ω, f ∈ P }⊂ ω Clopen subbase: { G ( n , f , A n , f ) : n ∈ ω, f ∈ P . } If ∅ � = U ⊂ open X then G ( n , f , A ) ⊂ U for some n ∈ ω , f ∈ � , A U ∈ �A� Claim: pd ( X ) = ω . • U : X → τ be a NEA. • Then U ( p ) ⊃ G ( n p , f p , A p ) for all p ∈ X • there is g ∈ P s.t. f p < ∗ g for all p ∈ X . • R = { g ( n ) : n ∈ ω } × ω pins down U • Let p ∈ X . Then U ( p ) ⊃ G ( n p , f p , A p ) . •

• X = � ω ω × ω, τ � • X n = ( ω n \ ω n − 1 ) × ω . • P = � ( ω n \ ω n − 1 ) . � � �� � If n ∈ ω , f ∈ P , A ⊂ ω let G ( n , f , A )= ω m \ f ( m ) × A . m ≥ n � ω . � Fix an independent family A = { A n , f : n ∈ ω, f ∈ P }⊂ ω Clopen subbase: { G ( n , f , A n , f ) : n ∈ ω, f ∈ P . } If ∅ � = U ⊂ open X then G ( n , f , A ) ⊂ U for some n ∈ ω , f ∈ � , A U ∈ �A� Claim: pd ( X ) = ω . • U : X → τ be a NEA. • Then U ( p ) ⊃ G ( n p , f p , A p ) for all p ∈ X • there is g ∈ P s.t. f p < ∗ g for all p ∈ X . • R = { g ( n ) : n ∈ ω } × ω pins down U • Let p ∈ X . Then U ( p ) ⊃ G ( n p , f p , A p ) . • ∃ n ≥ n p s.t. f p ( n ) < g ( n ) •

• X = � ω ω × ω, τ � • X n = ( ω n \ ω n − 1 ) × ω . • P = � ( ω n \ ω n − 1 ) . � � �� � If n ∈ ω , f ∈ P , A ⊂ ω let G ( n , f , A )= ω m \ f ( m ) × A . m ≥ n � ω . � Fix an independent family A = { A n , f : n ∈ ω, f ∈ P }⊂ ω Clopen subbase: { G ( n , f , A n , f ) : n ∈ ω, f ∈ P . } If ∅ � = U ⊂ open X then G ( n , f , A ) ⊂ U for some n ∈ ω , f ∈ � , A U ∈ �A� Claim: pd ( X ) = ω . X n ω n • U : X → τ be a NEA. • Then U ( p ) ⊃ G ( n p , f p , A p ) for all p ∈ X • there is g ∈ P s.t. f p < ∗ g for all p ∈ X . • R = { g ( n ) : n ∈ ω } × ω pins down U • Let p ∈ X . Then U ( p ) ⊃ G ( n p , f p , A p ) . • ∃ n ≥ n p s.t. f p ( n ) < g ( n ) •

• X = � ω ω × ω, τ � • X n = ( ω n \ ω n − 1 ) × ω . • P = � ( ω n \ ω n − 1 ) . � � �� � If n ∈ ω , f ∈ P , A ⊂ ω let G ( n , f , A )= ω m \ f ( m ) × A . m ≥ n � ω . � Fix an independent family A = { A n , f : n ∈ ω, f ∈ P }⊂ ω Clopen subbase: { G ( n , f , A n , f ) : n ∈ ω, f ∈ P . } If ∅ � = U ⊂ open X then G ( n , f , A ) ⊂ U for some n ∈ ω , f ∈ � , A U ∈ �A� Claim: pd ( X ) = ω . X n ω n • U : X → τ be a NEA. • Then U ( p ) ⊃ G ( n p , f p , A p ) for all p ∈ X • there is g ∈ P s.t. f p < ∗ g for all p ∈ X . f p ( n ) • R = { g ( n ) : n ∈ ω } × ω pins down U A p • Let p ∈ X . Then U ( p ) ⊃ G ( n p , f p , A p ) . • ∃ n ≥ n p s.t. f p ( n ) < g ( n ) •

• X = � ω ω × ω, τ � • X n = ( ω n \ ω n − 1 ) × ω . • P = � ( ω n \ ω n − 1 ) . � � �� � If n ∈ ω , f ∈ P , A ⊂ ω let G ( n , f , A )= ω m \ f ( m ) × A . m ≥ n � ω . � Fix an independent family A = { A n , f : n ∈ ω, f ∈ P }⊂ ω Clopen subbase: { G ( n , f , A n , f ) : n ∈ ω, f ∈ P . } If ∅ � = U ⊂ open X then G ( n , f , A ) ⊂ U for some n ∈ ω , f ∈ � , A U ∈ �A� Claim: pd ( X ) = ω . X n ω n • U : X → τ be a NEA. • Then U ( p ) ⊃ G ( n p , f p , A p ) for all p ∈ X • there is g ∈ P s.t. f p < ∗ g for all p ∈ X . U ( p ) ⊃ f p ( n ) • R = { g ( n ) : n ∈ ω } × ω pins down U A p • Let p ∈ X . Then U ( p ) ⊃ G ( n p , f p , A p ) . • ∃ n ≥ n p s.t. f p ( n ) < g ( n ) •

• X = � ω ω × ω, τ � • X n = ( ω n \ ω n − 1 ) × ω . • P = � ( ω n \ ω n − 1 ) . � � �� � If n ∈ ω , f ∈ P , A ⊂ ω let G ( n , f , A )= ω m \ f ( m ) × A . m ≥ n � ω . � Fix an independent family A = { A n , f : n ∈ ω, f ∈ P }⊂ ω Clopen subbase: { G ( n , f , A n , f ) : n ∈ ω, f ∈ P . } If ∅ � = U ⊂ open X then G ( n , f , A ) ⊂ U for some n ∈ ω , f ∈ � , A U ∈ �A� Claim: pd ( X ) = ω . X n ω n • U : X → τ be a NEA. • Then U ( p ) ⊃ G ( n p , f p , A p ) for all p ∈ X g ( n ) • there is g ∈ P s.t. f p < ∗ g for all p ∈ X . U ( p ) ⊃ f p ( n ) • R = { g ( n ) : n ∈ ω } × ω pins down U A p • Let p ∈ X . Then U ( p ) ⊃ G ( n p , f p , A p ) . • ∃ n ≥ n p s.t. f p ( n ) < g ( n ) •

• X = � ω ω × ω, τ � • X n = ( ω n \ ω n − 1 ) × ω . • P = � ( ω n \ ω n − 1 ) . � � �� � If n ∈ ω , f ∈ P , A ⊂ ω let G ( n , f , A )= ω m \ f ( m ) × A . m ≥ n � ω . � Fix an independent family A = { A n , f : n ∈ ω, f ∈ P }⊂ ω Clopen subbase: { G ( n , f , A n , f ) : n ∈ ω, f ∈ P . } If ∅ � = U ⊂ open X then G ( n , f , A ) ⊂ U for some n ∈ ω , f ∈ � , A U ∈ �A� Claim: pd ( X ) = ω . X n ω n • U : X → τ be a NEA. • Then U ( p ) ⊃ G ( n p , f p , A p ) for all p ∈ X g ( n ) ⊂ R ω • there is g ∈ P s.t. f p < ∗ g for all p ∈ X . U ( p ) ⊃ f p ( n ) • R = { g ( n ) : n ∈ ω } × ω pins down U A p • Let p ∈ X . Then U ( p ) ⊃ G ( n p , f p , A p ) . • ∃ n ≥ n p s.t. f p ( n ) < g ( n ) •

• X = � ω ω × ω, τ � • X n = ( ω n \ ω n − 1 ) × ω . • P = � ( ω n \ ω n − 1 ) . � � �� � If n ∈ ω , f ∈ P , A ⊂ ω let G ( n , f , A )= ω m \ f ( m ) × A . m ≥ n � ω . � Fix an independent family A = { A n , f : n ∈ ω, f ∈ P }⊂ ω Clopen subbase: { G ( n , f , A n , f ) : n ∈ ω, f ∈ P . } If ∅ � = U ⊂ open X then G ( n , f , A ) ⊂ U for some n ∈ ω , f ∈ � , A U ∈ �A� Claim: pd ( X ) = ω . X n ω n • U : X → τ be a NEA. • Then U ( p ) ⊃ G ( n p , f p , A p ) for all p ∈ X g ( n ) ⊂ R ω • there is g ∈ P s.t. f p < ∗ g for all p ∈ X . U ( p ) ⊃ f p ( n ) • R = { g ( n ) : n ∈ ω } × ω pins down U A p • Let p ∈ X . Then U ( p ) ⊃ G ( n p , f p , A p ) . • ∃ n ≥ n p s.t. f p ( n ) < g ( n ) • Then R ∩ X n ∩ G ( n p , f p , A p ) � = ∅ . •

Some observations

Some observations If pd ( X ) < d ( X ) , then ∃ Y ⊂ open X s.t. pd ( Y ) < d ( Y ) and ∆( Y ) = | Y | .

Some observations If pd ( X ) < d ( X ) , then ∃ Y ⊂ open X s.t. pd ( Y ) < d ( Y ) and ∆( Y ) = | Y | . First examples: pd ( X ) = cf ( | X | ) < d ( X ) = ∆( X ) = | X | .

Some observations If pd ( X ) < d ( X ) , then ∃ Y ⊂ open X s.t. pd ( Y ) < d ( Y ) and ∆( Y ) = | Y | . First examples: pd ( X ) = cf ( | X | ) < d ( X ) = ∆( X ) = | X | . Questions • Can d ( X ) be a regular cardinal? • Can | X | be a regular cardinal?

Some observations If pd ( X ) < d ( X ) , then ∃ Y ⊂ open X s.t. pd ( Y ) < d ( Y ) and ∆( Y ) = | Y | . First examples: pd ( X ) = cf ( | X | ) < d ( X ) = ∆( X ) = | X | . Questions • Can d ( X ) be a regular cardinal? • Can | X | be a regular cardinal? Modified construction: pd ( X ) = cf ( | X | ) < d ( X ) = cf ( d ( X )) < ∆( X ) = | X |

Shelah’s Strong Hypothesis

Shelah’s Strong Hypothesis • µ > cf ( µ )

Shelah’s Strong Hypothesis • µ > cf ( µ ) • S ( µ ) = { a ∈ [ µ ∩ Reg ] cf ( µ ) : sup a = µ }

Shelah’s Strong Hypothesis • µ > cf ( µ ) • S ( µ ) = { a ∈ [ µ ∩ Reg ] cf ( µ ) : sup a = µ } • U ( a ) = { D : D is an ultrafilter on a , D ∩ J bd [ a ] = ∅} .

Shelah’s Strong Hypothesis • µ > cf ( µ ) • S ( µ ) = { a ∈ [ µ ∩ Reg ] cf ( µ ) : sup a = µ } • U ( a ) = { D : D is an ultrafilter on a , D ∩ J bd [ a ] = ∅} . • pp ( µ )= sup { cf ( � a / D ) : a ∈ S ( µ ) , D ∈ U ( a )) }

Shelah’s Strong Hypothesis • µ > cf ( µ ) • S ( µ ) = { a ∈ [ µ ∩ Reg ] cf ( µ ) : sup a = µ } • U ( a ) = { D : D is an ultrafilter on a , D ∩ J bd [ a ] = ∅} . • pp ( µ )= sup { cf ( � a / D ) : a ∈ S ( µ ) , D ∈ U ( a )) } Shelah’s Strong Hypothesis: pp ( µ ) = µ + for all singular cardinal µ .

An equiconsistency result

An equiconsistency result Theorem (I. Juhász, L.S., Z. Szentmiklóssy) The following three statements are equiconsistent : (i) There is a singular cardinal λ with pp ( λ ) > λ + , i.e. Shelah’s Strong Hypothesis fails; (ii) there is a 0-dimensional Hausdorff space X such that | X | = ∆( X ) is a regular cardinal and pd ( X ) < d ( X ) ; (iii) there is a topological space X such that | X | = ∆( X ) is a regular cardinal and pd ( X ) < d ( X ) .

An equiconsistency result Theorem (I. Juhász, L.S., Z. Szentmiklóssy) The following three statements are equiconsistent : (i) There is a singular cardinal λ with pp ( λ ) > λ + , i.e. Shelah’s Strong Hypothesis fails; (ii) there is a 0-dimensional Hausdorff space X such that | X | = ∆( X ) is a regular cardinal and pd ( X ) < d ( X ) ; (iii) there is a topological space X such that | X | = ∆( X ) is a regular cardinal and pd ( X ) < d ( X ) . No equivalence: Con(failure of SSH + the limit cardinals are strong limit)

Connected and locally connected spaces

Connected and locally connected spaces Theorem (I. Juhász,J. van Mill, L.S., Z. Szentmiklóssy ) T:F.A.E: (1) 2 κ < κ + ω for each cardinal κ , (2) pd ( X ) = d ( X ) for each T 2 space X, (3) pd ( X ) = d ( X ) for each 0-dimensional T 2 space X.

Connected and locally connected spaces Theorem (I. Juhász,J. van Mill, L.S., Z. Szentmiklóssy ) T:F.A.E: (1) 2 κ < κ + ω for each cardinal κ , (2) pd ( X ) = d ( X ) for each T 2 space X, (3) pd ( X ) = d ( X ) for each 0-dimensional T 2 space X. (4) pd ( X ) = d ( X ) for all connected, locally connected regular spaces. (5) pd ( X ) = d ( X ) for all Abelian topological groups.

Connected and locally connected spaces Theorem (I. Juhász,J. van Mill, L.S., Z. Szentmiklóssy ) T:F.A.E: (1) 2 κ < κ + ω for each cardinal κ , (2) pd ( X ) = d ( X ) for each T 2 space X, (3) pd ( X ) = d ( X ) for each 0-dimensional T 2 space X. (4) pd ( X ) = d ( X ) for all connected, locally connected regular spaces. (5) pd ( X ) = d ( X ) for all Abelian topological groups. What about connected Tychonoff spaces?

Connected and locally connected spaces Theorem (I. Juhász,J. van Mill, L.S., Z. Szentmiklóssy ) T:F.A.E: (1) 2 κ < κ + ω for each cardinal κ , (2) pd ( X ) = d ( X ) for each T 2 space X, (3) pd ( X ) = d ( X ) for each 0-dimensional T 2 space X. (4) pd ( X ) = d ( X ) for all connected, locally connected regular spaces. (5) pd ( X ) = d ( X ) for all Abelian topological groups. What about connected Tychonoff spaces? Theorem (JvMSSz) It is consistent that

Connected and locally connected spaces Theorem (I. Juhász,J. van Mill, L.S., Z. Szentmiklóssy ) T:F.A.E: (1) 2 κ < κ + ω for each cardinal κ , (2) pd ( X ) = d ( X ) for each T 2 space X, (3) pd ( X ) = d ( X ) for each 0-dimensional T 2 space X. (4) pd ( X ) = d ( X ) for all connected, locally connected regular spaces. (5) pd ( X ) = d ( X ) for all Abelian topological groups. What about connected Tychonoff spaces? Theorem (JvMSSz) It is consistent that • there is a 0-dimensional space X with pd ( X ) < d ( X )

Connected and locally connected spaces Theorem (I. Juhász,J. van Mill, L.S., Z. Szentmiklóssy ) T:F.A.E: (1) 2 κ < κ + ω for each cardinal κ , (2) pd ( X ) = d ( X ) for each T 2 space X, (3) pd ( X ) = d ( X ) for each 0-dimensional T 2 space X. (4) pd ( X ) = d ( X ) for all connected, locally connected regular spaces. (5) pd ( X ) = d ( X ) for all Abelian topological groups. What about connected Tychonoff spaces? Theorem (JvMSSz) It is consistent that • there is a 0-dimensional space X with pd ( X ) < d ( X ) • pd ( X ) = d ( X ) for all connected Tychonoff spaces.

A connected, locally connected Tychonoff example

A connected, locally connected Tychonoff example If X is a connected, Tychonoff space then | X | ≥ 2 ω .

A connected, locally connected Tychonoff example If X is a connected, Tychonoff space then | X | ≥ 2 ω . Theorem (I. Juhász,J. van Mill, L.S., Z. Szentmiklóssy) T:F.A.E: (1) There is a singular cardinal µ ≥ 2 ω which is not a strong limit cardinal. (2) There is a connected, locally connected Tychonoff space X with ∆( X ) = | X | and pd ( X ) < d ( X ) .

A connected, locally connected Tychonoff example If X is a connected, Tychonoff space then | X | ≥ 2 ω . Theorem (I. Juhász,J. van Mill, L.S., Z. Szentmiklóssy) T:F.A.E: (1) There is a singular cardinal µ ≥ 2 ω which is not a strong limit cardinal. (2) There is a connected, locally connected Tychonoff space X with ∆( X ) = | X | and pd ( X ) < d ( X ) . (3) There is a pathwise connected, locally pathwise connected Tychonoff Abelian topological group X with ∆( X ) = | X | and pd ( X ) < d ( X ) .

Embedding theorem

Embedding theorem • µ is a singular cardinal µ ≥ 2 ω , not a strong limit cardinal.

Embedding theorem • µ is a singular cardinal µ ≥ 2 ω , not a strong limit cardinal. • Need pathwise connected, locally pathwise connected Tychonoff Abelian topological group H with ∆( H ) = | H | and pd ( H ) < d ( H ) .

Embedding theorem • µ is a singular cardinal µ ≥ 2 ω , not a strong limit cardinal. • Need pathwise connected, locally pathwise connected Tychonoff Abelian topological group H with ∆( H ) = | H | and pd ( H ) < d ( H ) . • there is a 0-dimensional neat space X such that pd ( X ) < d ( X ) and | X | ≥ 2 ω .

Embedding theorem • µ is a singular cardinal µ ≥ 2 ω , not a strong limit cardinal. • Need pathwise connected, locally pathwise connected Tychonoff Abelian topological group H with ∆( H ) = | H | and pd ( H ) < d ( H ) . • there is a 0-dimensional neat space X such that pd ( X ) < d ( X ) and | X | ≥ 2 ω . Theorem Let X be a T 3 . 5 neat space such that | X | ≥ 2 ω . Then X has a closed embedding into a T 3 . 5 Abelian topological group H such that

Embedding theorem • µ is a singular cardinal µ ≥ 2 ω , not a strong limit cardinal. • Need pathwise connected, locally pathwise connected Tychonoff Abelian topological group H with ∆( H ) = | H | and pd ( H ) < d ( H ) . • there is a 0-dimensional neat space X such that pd ( X ) < d ( X ) and | X | ≥ 2 ω . Theorem Let X be a T 3 . 5 neat space such that | X | ≥ 2 ω . Then X has a closed embedding into a T 3 . 5 Abelian topological group H such that 1. d ( X ) = d ( H ) , 2. pd ( X ) = pd ( H ) , 3. H is neat, 4. H is pathwise connected and locally pathwise connected.

Step 1: embedding into a group

Step 1: embedding into a group If X is a T 3 . 5 -space, then F ( X ) and A ( X ) denote the free topological group and the free abelian topological group on X .

Step 1: embedding into a group If X is a T 3 . 5 -space, then F ( X ) and A ( X ) denote the free topological group and the free abelian topological group on X . F ( X ) is a topological group containing (a homeomorphic copy of) X such that 1. X generates F ( X ) algebraically, 2. every continuous function f : X → H , where H is any topological group, can be extended to a continuous homomorphism ¯ f : F ( X ) → H .

Step 1: embedding into a group If X is a T 3 . 5 -space, then F ( X ) and A ( X ) denote the free topological group and the free abelian topological group on X . F ( X ) is a topological group containing (a homeomorphic copy of) X such that 1. X generates F ( X ) algebraically, 2. every continuous function f : X → H , where H is any topological group, can be extended to a continuous homomorphism ¯ f : F ( X ) → H . Similarly for A ( X ) .

Step 1: embedding into a group If X is a T 3 . 5 -space, then F ( X ) and A ( X ) denote the free topological group and the free abelian topological group on X . F ( X ) is a topological group containing (a homeomorphic copy of) X such that 1. X generates F ( X ) algebraically, 2. every continuous function f : X → H , where H is any topological group, can be extended to a continuous homomorphism ¯ f : F ( X ) → H . Similarly for A ( X ) . The existence of these groups was proved by Markov.

Step 1: embedding into a group If X is a T 3 . 5 -space, then F ( X ) and A ( X ) denote the free topological group and the free abelian topological group on X . F ( X ) is a topological group containing (a homeomorphic copy of) X such that 1. X generates F ( X ) algebraically, 2. every continuous function f : X → H , where H is any topological group, can be extended to a continuous homomorphism ¯ f : F ( X ) → H . Similarly for A ( X ) . The existence of these groups was proved by Markov. Theorem (JvMSSz) Let X be a T 3 . 5 -space. Then d ( X ) = d ( F ( X )) = d ( A ( X )) . If X is neat, then so are A ( X ) and F ( X ) , and pd ( X ) = pd ( A ( X )) = pd ( F ( X )) .

Step 2: embedding a group into a pathwise connected one

Step 2: embedding a group into a pathwise connected one • Hartman Mycielski construction

Step 2: embedding a group into a pathwise connected one • Hartman Mycielski construction • Let ( G , · , e ) be a Tychonoff topological group. G • = f ∈ [ 0 , 1 ) G : � for some sequence 0 = a 0 < a 1 < · · · < a n = 1 � f is constant on [ a k , a k + 1 ) for every k = 0 , . . . , n − 1 .

Step 2: embedding a group into a pathwise connected one • Hartman Mycielski construction • Let ( G , · , e ) be a Tychonoff topological group. G • = f ∈ [ 0 , 1 ) G : � for some sequence 0 = a 0 < a 1 < · · · < a n = 1 � f is constant on [ a k , a k + 1 ) for every k = 0 , . . . , n − 1 . • Define ∗ on G • by ( f ∗ g )( x ) = f ( x ) · g ( x ) for all f , g ∈ G • and x ∈ [ 0 , 1 ) .