On some topological properties of pointfree function rings Mark - PowerPoint PPT Presentation

Introduction Spatial setting Frames and L ( R ) ’Topologies’ on R ( L ) and R ∗ ( L ) Dini properties The S-W property Properties of R ( L ) Fact R ( L ) is a strong unital archimedean f -ring. ◮ l -ring: ◮ ( α ⋄ β ) + γ = ( α + γ ) ⋄ ( β + γ ) for ⋄ ∈ {∨ , ∧} ◮ αβ ≥ 0 if α, β ≥ 0 ◮ unital: unit is 1

Introduction Spatial setting Frames and L ( R ) ’Topologies’ on R ( L ) and R ∗ ( L ) Dini properties The S-W property Properties of R ( L ) Fact R ( L ) is a strong unital archimedean f -ring. ◮ l -ring: ◮ ( α ⋄ β ) + γ = ( α + γ ) ⋄ ( β + γ ) for ⋄ ∈ {∨ , ∧} ◮ αβ ≥ 0 if α, β ≥ 0 ◮ unital: unit is 1 ◮ strong: every α with α ≥ 1 is invertible

Introduction Spatial setting Frames and L ( R ) ’Topologies’ on R ( L ) and R ∗ ( L ) Dini properties The S-W property Properties of R ( L ) Fact R ( L ) is a strong unital archimedean f -ring. ◮ l -ring: ◮ ( α ⋄ β ) + γ = ( α + γ ) ⋄ ( β + γ ) for ⋄ ∈ {∨ , ∧} ◮ αβ ≥ 0 if α, β ≥ 0 ◮ unital: unit is 1 ◮ strong: every α with α ≥ 1 is invertible ◮ archimedean: α, β ≥ 0 and n α ≤ β (all n ) imply α = 0

Introduction Spatial setting Frames and L ( R ) ’Topologies’ on R ( L ) and R ∗ ( L ) Dini properties The S-W property Properties of R ( L ) Fact R ( L ) is a strong unital archimedean f -ring. ◮ l -ring: ◮ ( α ⋄ β ) + γ = ( α + γ ) ⋄ ( β + γ ) for ⋄ ∈ {∨ , ∧} ◮ αβ ≥ 0 if α, β ≥ 0 ◮ unital: unit is 1 ◮ strong: every α with α ≥ 1 is invertible ◮ archimedean: α, β ≥ 0 and n α ≤ β (all n ) imply α = 0 ◮ f -ring: | αβ | = | α || β | , with | α | := α ∨ ( − α )

Introduction Spatial setting Frames and L ( R ) ’Topologies’ on R ( L ) and R ∗ ( L ) Dini properties The S-W property The pointfree counterpart to C ∗ ( X ) For a frame L , let R ∗ ( L ) := { α ∈ R ( L ) | | α | ≤ n , some n } Fact R ∗ ( L ) is an l -subring of R ( L ) and hence also a strong unital archimedean f -ring.

Introduction Spatial setting Frames and L ( R ) ’Topologies’ on R ( L ) and R ∗ ( L ) Dini properties The S-W property The uniform topology: pointfree case

Introduction Spatial setting Frames and L ( R ) ’Topologies’ on R ( L ) and R ∗ ( L ) Dini properties The S-W property The uniform topology: pointfree case Definition For a frame L the uniform topology on R ( L ) is the topology having V n ( α ) := { γ ∈ R ( L ) | | α − γ | < 1 n } , all n ∈ N 0 as a base for the neighborhoods of α ∈ R ( L ).

Introduction Spatial setting Frames and L ( R ) ’Topologies’ on R ( L ) and R ∗ ( L ) Dini properties The S-W property The uniform topology: pointfree case Definition For a frame L the uniform topology on R ( L ) is the topology having V n ( α ) := { γ ∈ R ( L ) | | α − γ | < 1 n } , all n ∈ N 0 as a base for the neighborhoods of α ∈ R ( L ). Definition For a frame L the uniform topology on R ∗ ( L ) is the subspace topology it inherits from the uniform topology on R ( L ).

Introduction Spatial setting Frames and L ( R ) ’Topologies’ on R ( L ) and R ∗ ( L ) Dini properties The S-W property Pointwise convergence = convergence everywhere: spatial case Definition For a topological space X , a net ( f η ) η ∈ D and f ∈ C ( X ) we say that ( f η ) η ∈ D converges to f everywhere , and write ( f η ) η ∈ D → f , if ∀ x ∈ X , ∀ m ∈ N 0 , ∃ η 0 ∈ D , ∀ η ∈ D : η ≥ η 0 ⇒ | f ( x ) − f η ( x ) | < 1 m

Introduction Spatial setting Frames and L ( R ) ’Topologies’ on R ( L ) and R ∗ ( L ) Dini properties The S-W property Pointwise convergence = convergence everywhere: spatial case Definition For a topological space X , a net ( f η ) η ∈ D and f ∈ C ( X ) we say that ( f η ) η ∈ D converges to f everywhere , and write ( f η ) η ∈ D → f , if ∀ x ∈ X , ∀ m ∈ N 0 , ∃ η 0 ∈ D , ∀ η ∈ D : η ≥ η 0 ⇒ | f ( x ) − f η ( x ) | < 1 m Definition ◮ A net ( f η ) η ∈ D is called increasing if ∀ η, µ ∈ D : η ≤ µ ⇒ f η ≤ f µ . ◮ A net ( f η ) η ∈ D is called decreasing if ∀ η, µ ∈ D : η ≤ µ ⇒ f η ≥ f µ

Introduction Spatial setting Frames and L ( R ) ’Topologies’ on R ( L ) and R ∗ ( L ) Dini properties The S-W property Convergence everywhere for increasing/decreasing nets: spatial case So for ( f η ) η ∈ D increasing and with f η ≤ f for all η ∈ D : ( f η ) η ∈ D → f { x ∈ X | | f ( x ) − f η ( x ) | < 1 � � ⇔∀ m ∈ N 0 : m } = X η 0 ∈ D η ∈ D ,η ≥ η 0 { x ∈ X | f ( x ) − f η 0 ( x ) < 1 � ⇔∀ m ∈ N 0 : m } = X η 0 ∈ D � ⇔∀ m ∈ N 0 : { x ∈ X | (1 − m ( f ( x ) − f η 0 ( x ))) > 0 } = X η 0 ∈ D � ⇔∀ m ∈ N 0 : { x ∈ X | (1 − m ( f ( x ) − f η 0 ( x ))) ∨ 0 � = 0 } = X η 0 ∈ D

Introduction Spatial setting Frames and L ( R ) ’Topologies’ on R ( L ) and R ∗ ( L ) Dini properties The S-W property The cozero part of a frame - completely regular frames Notation: in L ( R ), for every p ∈ Q � ( − , p ) := { ( q , p ) | q ∈ Q , q < p } � ( p , − ) := { ( p , q ) | q ∈ Q , q > p }

Introduction Spatial setting Frames and L ( R ) ’Topologies’ on R ( L ) and R ∗ ( L ) Dini properties The S-W property The cozero part of a frame - completely regular frames Notation: in L ( R ), for every p ∈ Q � ( − , p ) := { ( q , p ) | q ∈ Q , q < p } � ( p , − ) := { ( p , q ) | q ∈ Q , q > p } Definition For a frame L and α ∈ R ( L ), coz( α ) := α (( − , 0) ∨ (0 , − )) is called the cozero element determined by α . Coz L := { coz( α ) | α ∈ R ( L ) } is called the cozero part of L .

Introduction Spatial setting Frames and L ( R ) ’Topologies’ on R ( L ) and R ∗ ( L ) Dini properties The S-W property The cozero part of a frame - completely regular frames Fact For any frame L , Coz( L ) is a sub- σ -frame of L . Definition L a frame, a , b ∈ L : ◮ a ≺ b ( a rather below b ) ≡ a ∗ ∨ b = e ◮ a ≺≺ b ( a well below b ) ≡ exists ( a r ) r ∈ D such that a 0 = a , a 1 = b , and a r ≺ a s wehenver r < s ◮ L completely regular ≡ a = � { x ∈ L | x ≺≺ a } for all a ∈ L Fact A frame L is completely regular if and only if it is ( � )-generated by Coz L .

Introduction Spatial setting Frames and L ( R ) ’Topologies’ on R ( L ) and R ∗ ( L ) Dini properties The S-W property Convergence everywhere for increasing nets: pointfree case Definition Let L be a frame, α ∈ R ( L ) and ( α η ) η ∈ D a net in R ( L ). Then we say that ( α η ) η ∈ D increases everywhere to α , and we write ( α η ) η ∈ D ↑ α if ( α η ) η ∈ D is increasing, α η ≤ α for all η ∈ D , and � coz(( 1 − m ( α − α η )) + ) = e . ∀ m ∈ N 0 : η ∈ D Notation: for γ ∈ R ( L ), we write γ + := γ ∨ 0

Introduction Spatial setting Frames and L ( R ) ’Topologies’ on R ( L ) and R ∗ ( L ) Dini properties The S-W property Convergence everywhere for decreasing nets: pointfree case Definition Let L be a frame, α ∈ R ( L ) and ( α η ) η ∈ D a net in R ( L ). Then we say that ( α η ) η decreases everywhere to α , and we write ( α η ) η ∈ D ↓ α if ( α η ) η ∈ D is decreasing, α η ≥ α for all η ∈ D , and � coz(( 1 − m ( α η − α )) + ) = e . ∀ m ∈ N 0 : η ∈ D Notation: for γ ∈ R ( L ), we write γ + := γ ∨ 0

Introduction Spatial setting Frames and L ( R ) ’Topologies’ on R ( L ) and R ∗ ( L ) Dini properties The S-W property The weak Dini Property Definition (wDP) For a frame L , we say that L satisfies the weak Dini property or (wDP) if for any α ∈ R ( L ) and any sequence ( α n ) n in R ( L ) which increases everywhere to α , the sequence ( α n ) n converges to α in the uniform topology on R ( L ).

Introduction Spatial setting Frames and L ( R ) ’Topologies’ on R ( L ) and R ∗ ( L ) Dini properties The S-W property The weak Dini Property Definition (wDP) For a frame L , we say that L satisfies the weak Dini property or (wDP) if for any α ∈ R ( L ) and any sequence ( α n ) n in R ( L ) which increases everywhere to α , the sequence ( α n ) n converges to α in the uniform topology on R ( L ). Remark: note that (wDP) is equivalent to the statement with ‘increasing’ → ‘decreasing’

Introduction Spatial setting Frames and L ( R ) ’Topologies’ on R ( L ) and R ∗ ( L ) Dini properties The S-W property Pointfree pseudo-compactness Definition A frame L is called pseudo-compact if every element of R ( L ) is bounded, i.e. if R ( L ) = R ∗ ( L ).

Introduction Spatial setting Frames and L ( R ) ’Topologies’ on R ( L ) and R ∗ ( L ) Dini properties The S-W property Pointfree pseudo-compactness Definition A frame L is called pseudo-compact if every element of R ( L ) is bounded, i.e. if R ( L ) = R ∗ ( L ). Theorem (Banaschewski-Gilmour) For any frame L , the following are equivalent: (1) L is pseudo-compact. (2) Any sequence a 0 ≺≺ a 1 ≺≺ a 2 ≺≺ . . . such that � a n = e in L terminates, that is, a k = e for some k . (3) The σ -frame Coz L is compact.

Introduction Spatial setting Frames and L ( R ) ’Topologies’ on R ( L ) and R ∗ ( L ) Dini properties The S-W property Characterizing (wDP) Theorem For a frame L the following assertions are equivalent: (1) L satisfies (wDP). (2) L is pseudo-compact.

Introduction Spatial setting Frames and L ( R ) ’Topologies’ on R ( L ) and R ∗ ( L ) Dini properties The S-W property Characterizing (wDP) Theorem For a frame L the following assertions are equivalent: (1) L satisfies (wDP). (2) L is pseudo-compact. Proof:

Introduction Spatial setting Frames and L ( R ) ’Topologies’ on R ( L ) and R ∗ ( L ) Dini properties The S-W property Characterizing (wDP) Theorem For a frame L the following assertions are equivalent: (1) L satisfies (wDP). (2) L is pseudo-compact. Proof: (1) ⇒ (2):

Introduction Spatial setting Frames and L ( R ) ’Topologies’ on R ( L ) and R ∗ ( L ) Dini properties The S-W property Characterizing (wDP) Theorem For a frame L the following assertions are equivalent: (1) L satisfies (wDP). (2) L is pseudo-compact. Proof: (1) ⇒ (2): ◮ take α ∈ R ( L ), α ≥ 0

Introduction Spatial setting Frames and L ( R ) ’Topologies’ on R ( L ) and R ∗ ( L ) Dini properties The S-W property Characterizing (wDP) Theorem For a frame L the following assertions are equivalent: (1) L satisfies (wDP). (2) L is pseudo-compact. Proof: (1) ⇒ (2): ◮ take α ∈ R ( L ), α ≥ 0 ◮ show that ( α ∧ n ) n ↑ α

Introduction Spatial setting Frames and L ( R ) ’Topologies’ on R ( L ) and R ∗ ( L ) Dini properties The S-W property Characterizing (wDP) Theorem For a frame L the following assertions are equivalent: (1) L satisfies (wDP). (2) L is pseudo-compact. Proof: (1) ⇒ (2): ◮ take α ∈ R ( L ), α ≥ 0 ◮ show that ( α ∧ n ) n ↑ α ◮ by (wDP), ( α ∧ n ) n converges to α w.r.t. the uniform topology

Introduction Spatial setting Frames and L ( R ) ’Topologies’ on R ( L ) and R ∗ ( L ) Dini properties The S-W property Characterizing (wDP) Theorem For a frame L the following assertions are equivalent: (1) L satisfies (wDP). (2) L is pseudo-compact. Proof: (1) ⇒ (2): ◮ take α ∈ R ( L ), α ≥ 0 ◮ show that ( α ∧ n ) n ↑ α ◮ by (wDP), ( α ∧ n ) n converges to α w.r.t. the uniform topology ◮ so α − α ∧ n ≤ 1 for some n , hence α is bounded

Introduction Spatial setting Frames and L ( R ) ’Topologies’ on R ( L ) and R ∗ ( L ) Dini properties The S-W property Characterizing (wDP) (2) ⇒ (1):

Introduction Spatial setting Frames and L ( R ) ’Topologies’ on R ( L ) and R ∗ ( L ) Dini properties The S-W property Characterizing (wDP) (2) ⇒ (1): ◮ assume ( α n ) n ↑ α

Introduction Spatial setting Frames and L ( R ) ’Topologies’ on R ( L ) and R ∗ ( L ) Dini properties The S-W property Characterizing (wDP) (2) ⇒ (1): ◮ assume ( α n ) n ↑ α ◮ ∀ m ∈ N 0 : � n ∈ N 0 coz(( 1 − m ( α − α n )) + ) = e

Introduction Spatial setting Frames and L ( R ) ’Topologies’ on R ( L ) and R ∗ ( L ) Dini properties The S-W property Characterizing (wDP) (2) ⇒ (1): ◮ assume ( α n ) n ↑ α ◮ ∀ m ∈ N 0 : � n ∈ N 0 coz(( 1 − m ( α − α n )) + ) = e ◮ invoking pseudo-compactness, form ( n m ) m such that ∀ m ∈ N 0 : coz(( 1 − m ( α − α n m )) + ) = e

Introduction Spatial setting Frames and L ( R ) ’Topologies’ on R ( L ) and R ∗ ( L ) Dini properties The S-W property Characterizing (wDP) (2) ⇒ (1): ◮ assume ( α n ) n ↑ α ◮ ∀ m ∈ N 0 : � n ∈ N 0 coz(( 1 − m ( α − α n )) + ) = e ◮ invoking pseudo-compactness, form ( n m ) m such that ∀ m ∈ N 0 : coz(( 1 − m ( α − α n m )) + ) = e ◮ then ∀ m ∈ N 0 : α − α n m ≤ 1 m

Introduction Spatial setting Frames and L ( R ) ’Topologies’ on R ( L ) and R ∗ ( L ) Dini properties The S-W property Characterizing (wDP) (2) ⇒ (1): ◮ assume ( α n ) n ↑ α ◮ ∀ m ∈ N 0 : � n ∈ N 0 coz(( 1 − m ( α − α n )) + ) = e ◮ invoking pseudo-compactness, form ( n m ) m such that ∀ m ∈ N 0 : coz(( 1 − m ( α − α n m )) + ) = e ◮ then ∀ m ∈ N 0 : α − α n m ≤ 1 m ◮ so, since ( α n ) n is increasing, ( α n ) n converges to α w.r.t. the uniform topology �

Introduction Spatial setting Frames and L ( R ) ’Topologies’ on R ( L ) and R ∗ ( L ) Dini properties The S-W property The Strong Dini Poperty Definition (sDP) For a frame L , we say that L satisfies the strong Dini property or (sDP) if for any α ∈ R L and any net ( α η ) η ∈ D in R L which increases everywhere to α , the net ( α η ) η ∈ D converges to α in the uniform topology on R L .

Introduction Spatial setting Frames and L ( R ) ’Topologies’ on R ( L ) and R ∗ ( L ) Dini properties The S-W property The Strong Dini Poperty Definition (sDP) For a frame L , we say that L satisfies the strong Dini property or (sDP) if for any α ∈ R L and any net ( α η ) η ∈ D in R L which increases everywhere to α , the net ( α η ) η ∈ D converges to α in the uniform topology on R L . Remark: note that (sDP) is equivalent to the statement with ‘increasing’ → ‘decreasing’

Introduction Spatial setting Frames and L ( R ) ’Topologies’ on R ( L ) and R ∗ ( L ) Dini properties The S-W property Characterizing (sDP) Theorem For a frame L the following assertions are equivalent: (1) L satisfies (sDP). (2) Every cover L consisting of cozero elements has a finite subcover. (3) The completely regular coreflection of L compact.

Introduction Spatial setting Frames and L ( R ) ’Topologies’ on R ( L ) and R ∗ ( L ) Dini properties The S-W property Characterizing (sDP) Theorem For a frame L the following assertions are equivalent: (1) L satisfies (sDP). (2) Every cover L consisting of cozero elements has a finite subcover. (3) The completely regular coreflection of L compact. Proof:

Introduction Spatial setting Frames and L ( R ) ’Topologies’ on R ( L ) and R ∗ ( L ) Dini properties The S-W property Characterizing (sDP) Theorem For a frame L the following assertions are equivalent: (1) L satisfies (sDP). (2) Every cover L consisting of cozero elements has a finite subcover. (3) The completely regular coreflection of L compact. Proof: (2) ⇔ (3): clear

Introduction Spatial setting Frames and L ( R ) ’Topologies’ on R ( L ) and R ∗ ( L ) Dini properties The S-W property Characterizing (sDP) (2) ⇒ (1):

Introduction Spatial setting Frames and L ( R ) ’Topologies’ on R ( L ) and R ∗ ( L ) Dini properties The S-W property Characterizing (sDP) (2) ⇒ (1): ◮ assume ( α η ) η ∈ D ↓ 0 in R ( L ), fix m ∈ N 0

Introduction Spatial setting Frames and L ( R ) ’Topologies’ on R ( L ) and R ∗ ( L ) Dini properties The S-W property Characterizing (sDP) (2) ⇒ (1): ◮ assume ( α η ) η ∈ D ↓ 0 in R ( L ), fix m ∈ N 0 ◮ � η ∈ D coz(( 1 − m α η ) + ) = e

Introduction Spatial setting Frames and L ( R ) ’Topologies’ on R ( L ) and R ∗ ( L ) Dini properties The S-W property Characterizing (sDP) (2) ⇒ (1): ◮ assume ( α η ) η ∈ D ↓ 0 in R ( L ), fix m ∈ N 0 ◮ � η ∈ D coz(( 1 − m α η ) + ) = e ◮ using (2), pick η 0 ∈ D with coz(( 1 − m α η 0 ) + ) = e

Introduction Spatial setting Frames and L ( R ) ’Topologies’ on R ( L ) and R ∗ ( L ) Dini properties The S-W property Characterizing (sDP) (2) ⇒ (1): ◮ assume ( α η ) η ∈ D ↓ 0 in R ( L ), fix m ∈ N 0 ◮ � η ∈ D coz(( 1 − m α η ) + ) = e ◮ using (2), pick η 0 ∈ D with coz(( 1 − m α η 0 ) + ) = e ◮ then α η 0 ≤ 1 m

Introduction Spatial setting Frames and L ( R ) ’Topologies’ on R ( L ) and R ∗ ( L ) Dini properties The S-W property Characterizing (sDP) (2) ⇒ (1): ◮ assume ( α η ) η ∈ D ↓ 0 in R ( L ), fix m ∈ N 0 ◮ � η ∈ D coz(( 1 − m α η ) + ) = e ◮ using (2), pick η 0 ∈ D with coz(( 1 − m α η 0 ) + ) = e ◮ then α η 0 ≤ 1 m ◮ remember ( α η ) η ∈ D is decreasing ◮ so ( α η ) η converges to α w.r.t. the uniform topology �

Introduction Spatial setting Frames and L ( R ) ’Topologies’ on R ( L ) and R ∗ ( L ) Dini properties The S-W property Characterizing (sDP) (1) ⇒ (2):

Introduction Spatial setting Frames and L ( R ) ’Topologies’ on R ( L ) and R ∗ ( L ) Dini properties The S-W property Characterizing (sDP) (1) ⇒ (2): ◮ take F ⊆ Coz( L ) with � F = e

Introduction Spatial setting Frames and L ( R ) ’Topologies’ on R ( L ) and R ∗ ( L ) Dini properties The S-W property Characterizing (sDP) (1) ⇒ (2): ◮ take F ⊆ Coz( L ) with � F = e ◮ for all a ∈ F , pick α a ∈ R ( L ) with 0 ≤ α ≤ 1 such that coz( α a ) = a

Introduction Spatial setting Frames and L ( R ) ’Topologies’ on R ( L ) and R ∗ ( L ) Dini properties The S-W property Characterizing (sDP) (1) ⇒ (2): ◮ take F ⊆ Coz( L ) with � F = e ◮ for all a ∈ F , pick α a ∈ R ( L ) with 0 ≤ α ≤ 1 such that coz( α a ) = a ◮ put D := P fin ( F × N 0 ), ordered by ⊆

Introduction Spatial setting Frames and L ( R ) ’Topologies’ on R ( L ) and R ∗ ( L ) Dini properties The S-W property Characterizing (sDP) (1) ⇒ (2): ◮ take F ⊆ Coz( L ) with � F = e ◮ for all a ∈ F , pick α a ∈ R ( L ) with 0 ≤ α ≤ 1 such that coz( α a ) = a ◮ put D := P fin ( F × N 0 ), ordered by ⊆ ◮ for all η ∈ D , define � ( n α a − 1 ) + ∧ 1 β η := ( a , n ) ∈ η

Introduction Spatial setting Frames and L ( R ) ’Topologies’ on R ( L ) and R ∗ ( L ) Dini properties The S-W property Characterizing (sDP) (1) ⇒ (2): ◮ take F ⊆ Coz( L ) with � F = e ◮ for all a ∈ F , pick α a ∈ R ( L ) with 0 ≤ α ≤ 1 such that coz( α a ) = a ◮ put D := P fin ( F × N 0 ), ordered by ⊆ ◮ for all η ∈ D , define � ( n α a − 1 ) + ∧ 1 β η := ( a , n ) ∈ η ◮ verify that ( β η ) η ∈ D ↑ 1

Introduction Spatial setting Frames and L ( R ) ’Topologies’ on R ( L ) and R ∗ ( L ) Dini properties The S-W property Characterizing (sDP) ◮ by (sDP), ( β η ) η ∈ D converges to 1 w.r.t. the uniform topology

Introduction Spatial setting Frames and L ( R ) ’Topologies’ on R ( L ) and R ∗ ( L ) Dini properties The S-W property Characterizing (sDP) ◮ by (sDP), ( β η ) η ∈ D converges to 1 w.r.t. the uniform topology ◮ pick η ∈ D ( η � = ∅ ) such that ( n α a − 1 ) + ∧ 1 ≥ 1 � β η = 2 ( a , n ) ∈ η

Introduction Spatial setting Frames and L ( R ) ’Topologies’ on R ( L ) and R ∗ ( L ) Dini properties The S-W property Characterizing (sDP) ◮ by (sDP), ( β η ) η ∈ D converges to 1 w.r.t. the uniform topology ◮ pick η ∈ D ( η � = ∅ ) such that ( n α a − 1 ) + ∧ 1 ≥ 1 � β η = 2 ( a , n ) ∈ η ◮ since R ( L ) is an l -ring, this implies that ( n α a − 1 ) + ≥ 1 � 2 ( a , n ) ∈ η

Introduction Spatial setting Frames and L ( R ) ’Topologies’ on R ( L ) and R ∗ ( L ) Dini properties The S-W property Characterizing (sDP) ◮ by (sDP), ( β η ) η ∈ D converges to 1 w.r.t. the uniform topology ◮ pick η ∈ D ( η � = ∅ ) such that ( n α a − 1 ) + ∧ 1 ≥ 1 � β η = 2 ( a , n ) ∈ η ◮ since R ( L ) is an l -ring, this implies that ( n α a − 1 ) + ≥ 1 � 2 ( a , n ) ∈ η ◮ so    � � � coz( n α a − 1 ) + ≤ ( n α a − 1 ) +  = e = coz a ( a , n ) ∈ η ( a , n ) ∈ η ( a , n ) ∈ η �

Introduction Spatial setting Frames and L ( R ) ’Topologies’ on R ( L ) and R ∗ ( L ) Dini properties The S-W property Variant on a theme: the κ -Dini Property

Introduction Spatial setting Frames and L ( R ) ’Topologies’ on R ( L ) and R ∗ ( L ) Dini properties The S-W property Variant on a theme: the κ -Dini Property Definition For an infinite cardinal number κ , a frame L is called initially κ -compact if every cover of L of cardinality at most κ admits a finite subcover.

Introduction Spatial setting Frames and L ( R ) ’Topologies’ on R ( L ) and R ∗ ( L ) Dini properties The S-W property Variant on a theme: the κ -Dini Property Definition For an infinite cardinal number κ , a frame L is called initially κ -compact if every cover of L of cardinality at most κ admits a finite subcover. Note: for κ = ℵ 0 , initially κ -compact means countably compact.

Introduction Spatial setting Frames and L ( R ) ’Topologies’ on R ( L ) and R ∗ ( L ) Dini properties The S-W property Variant on a theme: the κ -Dini Property Definition For an infinite cardinal number κ , a frame L is called initially κ -compact if every cover of L of cardinality at most κ admits a finite subcover. Note: for κ = ℵ 0 , initially κ -compact means countably compact. Definition ( κ -DP) For a frame L and an infinite cardinal number κ , we say that L satisfies the κ -Dini property or ( κ -DP) if for any α ∈ R ( L ) and any net ( α η ) η ∈ D in R ( L ) with cardinality of D at most κ and which increases everywhere to α , the net ( α η ) η ∈ D converges to α in the uniform topology on R ( L ).

Introduction Spatial setting Frames and L ( R ) ’Topologies’ on R ( L ) and R ∗ ( L ) Dini properties The S-W property Characterizing ( κ -DP) Corollary For a frame L and an infinite cardinal number κ , the following assertions are equivalent: (1) L satisfies ( κ -DP). (2) Every cover L consisting of cozero elements and of cardinality at most κ has a finite subcover. (3) The completely regular coreflection of L is initially κ -compact.

Introduction Spatial setting Frames and L ( R ) ’Topologies’ on R ( L ) and R ∗ ( L ) Dini properties The S-W property Characterizing ( κ -DP) Corollary For a frame L and an infinite cardinal number κ , the following assertions are equivalent: (1) L satisfies ( κ -DP). (2) Every cover L consisting of cozero elements and of cardinality at most κ has a finite subcover. (3) The completely regular coreflection of L is initially κ -compact. Proof:

Introduction Spatial setting Frames and L ( R ) ’Topologies’ on R ( L ) and R ∗ ( L ) Dini properties The S-W property Characterizing ( κ -DP) Corollary For a frame L and an infinite cardinal number κ , the following assertions are equivalent: (1) L satisfies ( κ -DP). (2) Every cover L consisting of cozero elements and of cardinality at most κ has a finite subcover. (3) The completely regular coreflection of L is initially κ -compact. Proof: Note that for κ infinite and Card( F ) ≤ κ : � ( F × N 0 ) n ) ≤ κ. Card( P fin ( F × N 0 )) = Card( n ∈ N �

Introduction Spatial setting Frames and L ( R ) ’Topologies’ on R ( L ) and R ∗ ( L ) Dini properties The S-W property Some terminology Definition ( κ an infinite cardinal number) A frame L is called ◮ Lindel¨ of if every cover of L admits a countable subcover ◮ quasi-Lindel¨ of if every cover of L consisting of cozero elements admits a countable subcover ◮ initially κ -Lindel¨ of , if every cover of L of cardinality at most κ admits a countable subcover ◮ initially κ -quasi-Lindel¨ of , if every cover of L consisting of cozero elements and of cardinality at most κ admits a countable subcover

Introduction Spatial setting Frames and L ( R ) ’Topologies’ on R ( L ) and R ∗ ( L ) Dini properties The S-W property Characterizing (sDP) and ( κ -DP) Proposition For a frame L , the following assertions are equivalent: (1) L satisfies (sDP). (2) L is quasi-Lindel¨ of and L satisfies (wDP).

Introduction Spatial setting Frames and L ( R ) ’Topologies’ on R ( L ) and R ∗ ( L ) Dini properties The S-W property Characterizing (sDP) and ( κ -DP) Proposition For a frame L , the following assertions are equivalent: (1) L satisfies (sDP). (2) L is quasi-Lindel¨ of and L satisfies (wDP). Proposition For a frame L and an infinite cardinal number κ , the following assertions are equivalent: (1) L satisfies ( κ -DP). (2) L is initially κ -quasi-Lindel¨ of and L satisfies (wDP).

Introduction Spatial setting Frames and L ( R ) ’Topologies’ on R ( L ) and R ∗ ( L ) Dini properties The S-W property Characterizing quasi-Lindel¨ ofness Theorem For a frame L , the following assertions are equivalent: (1) L is quasi-Lindel¨ of. (2) The completely regular coreflection of L is Lindel¨ of. (3) For any net ( α η ) η ∈ D in R ( L ) and any α ∈ R ( L ) such that ( α η ) η ∈ D ↑ α (resp. ( α η ) η ∈ D ↓ α ), there exists an increasing sequence ( η n ) n in D such that ( α η n ) n ↑ α (resp. ( α η n ) n ↓ α ).

Introduction Spatial setting Frames and L ( R ) ’Topologies’ on R ( L ) and R ∗ ( L ) Dini properties The S-W property Characterizing initially κ -quasi-Lindel¨ ofness Theorem For a frame L and an infinite cardinal number κ , the following assertions are equivalent: (1) L is initially κ -quasi-Lindel¨ of. (2) The completely regular coreflection of L is initially κ -Lindel¨ of. (3) For any net ( α η ) η ∈ D in R ( L ) with cardinality of D at most κ and any α ∈ R ( L ) such that ( α η ) η ∈ D ↑ α (resp. ( α η ) η ∈ D ↓ α ), there exists an increasing sequence ( η n ) n in D such that ( α η n ) n ↑ α (resp. ( α η n ) n ↓ α ).

Introduction Spatial setting Frames and L ( R ) ’Topologies’ on R ( L ) and R ∗ ( L ) Dini properties The S-W property A final characterization of (sDP), ( κ -DP) and (wDP) Definition ( κ an infinite cardinal number) A frame L is called ◮ almost-compact if for every cover F of L , there exists S ⊆ F finite such that � S ) ∗ = 0 . ( ◮ initially κ -almost-compact if for every cover F of L of cardinalty at most κ , there exists S ⊆ F finite such that � S ) ∗ = 0 . (

Introduction Spatial setting Frames and L ( R ) ’Topologies’ on R ( L ) and R ∗ ( L ) Dini properties The S-W property A final characterization of (sDP), ( κ -DP) and (wDP) Definition ( κ an infinite cardinal number) A frame L is called ◮ quasi-almost-compact if for every cover F of L consisting of cozero elements and of cardinalty at most κ , there exists S ⊆ F finite such that � S ) ∗ = 0 . ( ◮ initially κ -quasi-almost-compact if for every cover F of L consisting of cozero elements and of cardinalty at most κ , there exists S ⊆ F finite such that � S ) ∗ = 0 . (

Introduction Spatial setting Frames and L ( R ) ’Topologies’ on R ( L ) and R ∗ ( L ) Dini properties The S-W property A final characterization of (sDP), ( κ -DP) and (wDP) Proposition Every quasi-almost-compact frame satisfies (sDP). For any infinite cardinal number κ , every initially κ -quasi-almost-compact frame satisfies ( κ -DP).