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 pd-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 pd-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 pd-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 pd-example

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

A connected, locally connected Tychonoff pd-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 neat, connected, locally connected Tychonoff space X with singular ∆( X ) = | X | and pd ( X ) < d ( X ) .

A connected, locally connected Tychonoff pd-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 neat, connected, locally connected Tychonoff space X with singular ∆( X ) = | X | and pd ( X ) < d ( X ) . (3) There is a neat, pathwise connected, locally pathwise connected Tychonoff Abelian topological group X with singular ∆( X ) = | X | and pd ( X ) < d ( X ) .

Extension theorems

Extension theorems connected T 3 pd-example connected, locally connected T 3 pd-example 0-dimensional pd-example group pd-example locally pathwise connected T 3 . 5 group pd-example

Extension theorems connected T 3 pd-example (2) (1) connected, locally connected T 3 pd-example 0-dimensional pd-example (3) group pd-example (4) locally pathwise connected T 3 . 5 group pd-example

T 3 pd-example = ⇒ connected T 3 pd-example

T 3 pd-example = ⇒ connected T 3 pd-example • Assume that X is a T 3 pd-example.

T 3 pd-example = ⇒ connected T 3 pd-example • Assume that X is a T 3 pd-example. • Ciesielski and Wojciechowsk: there is a separable connected T 3 space P of size ω 1

T 3 pd-example = ⇒ connected T 3 pd-example • Assume that X is a T 3 pd-example. • Ciesielski and Wojciechowsk: there is a separable connected T 3 space P of size ω 1 • Fix p ∈ P . The underlying set of Z is � � X × ( P \ { p } ) ∪ {∞} .

T 3 pd-example = ⇒ connected T 3 pd-example • Assume that X is a T 3 pd-example. • Ciesielski and Wojciechowsk: there is a separable connected T 3 space P of size ω 1 • Fix p ∈ P . The underlying set of Z is � � X × ( P \ { p } ) ∪ {∞} . • Topology on X × ( P \ { p } ) in Z is the product topology. A basic neighborhood of ∞ has the form � � X × ( U \ { p } ) ∪ {∞} , where U is any neighborhood of p in P .

T 3 pd-example = ⇒ connected T 3 pd-example • Assume that X is a T 3 pd-example. • Ciesielski and Wojciechowsk: there is a separable connected T 3 space P of size ω 1 • Fix p ∈ P . The underlying set of Z is � � X × ( P \ { p } ) ∪ {∞} . • Topology on X × ( P \ { p } ) in Z is the product topology. A basic neighborhood of ∞ has the form � � X × ( U \ { p } ) ∪ {∞} , where U is any neighborhood of p in P . • Theorem: Z is connected T 3 , d ( X ) = d ( Z ) and pd ( X ) = pd ( Z ) .

connected T 3 pd-example = ⇒ connected, loc. connected T 3 pd-example

connected T 3 pd-example = ⇒ connected, loc. connected T 3 pd-example • de Groot introduced the superextension of X denoted by λ X

connected T 3 pd-example = ⇒ connected, loc. connected T 3 pd-example • de Groot introduced the superextension of X denoted by λ X • L is linked system if any two of its members meet.

connected T 3 pd-example = ⇒ connected, loc. connected T 3 pd-example • de Groot introduced the superextension of X denoted by λ X • L is linked system if any two of its members meet. • λ X = {L : L is a maximal linked family of of closed subsets of X . }

connected T 3 pd-example = ⇒ connected, loc. connected T 3 pd-example • de Groot introduced the superextension of X denoted by λ X • L is linked system if any two of its members meet. • λ X = {L : L is a maximal linked family of of closed subsets of X . } • For A ⊂ X let A + = {M ∈ λ X : ( ∃ M ∈ M )( M ⊂ A ) } .

connected T 3 pd-example = ⇒ connected, loc. connected T 3 pd-example • de Groot introduced the superextension of X denoted by λ X • L is linked system if any two of its members meet. • λ X = {L : L is a maximal linked family of of closed subsets of X . } • For A ⊂ X let A + = {M ∈ λ X : ( ∃ M ∈ M )( M ⊂ A ) } . • closed subbase of λ X : { A + : A is closed in X }

connected T 3 pd-example = ⇒ connected, loc. connected T 3 pd-example • de Groot introduced the superextension of X denoted by λ X • L is linked system if any two of its members meet. • λ X = {L : L is a maximal linked family of of closed subsets of X . } • For A ⊂ X let A + = {M ∈ λ X : ( ∃ M ∈ M )( M ⊂ A ) } . • closed subbase of λ X : { A + : A is closed in X } • λ f X = {L ∈ λ X : ∃ M ∈ [ X ] <ω ( ∀ L ∈ L ) L ∩ M ∈ L}