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Partial Continuous Functions and Admissible Domain Representations Fredrik Dahlgren ([email protected]) Department of mathematics at Uppsala university CiE 2006, 30 June – 5 July Representing topological spaces A domain representation D of a space X is a triple (D, DR , δ ) where Representing topological spaces A domain representation D of a space X is a triple (D, DR , δ ) where X Representing topological spaces A domain representation D of a space X is a triple (D, DR , δ ) where X • D is a domain. D Representing topological spaces A domain representation D of a space X is a triple (D, DR , δ ) where X • D is a domain. • DR is a subset of D. D Representing topological spaces A domain representation D of a space X is a triple (D, DR , δ ) where X • D is a domain. • DR is a subset of D. • δ : DR → X is continuous and onto. δ D Representing continuous functions Suppose that we have domain representations of two topological spaces X and Y. Representing continuous functions Suppose that we have domain representations of two topological spaces X and Y. X Y δ D ε E Representing continuous functions Suppose that we have domain representations of two topological spaces X and Y. X Y δ D ε E Which continuous functions from X to Y lift to continuous functions on the domain representations? Representing continuous functions Suppose that we have domain representations of two topological spaces X and Y. f X δ D Y ε E Which continuous functions from X to Y lift to continuous functions on the domain representations? Representing continuous functions Suppose that we have domain representations of two topological spaces X and Y. f X Y ε δ D E fˉ Which continuous functions from X to Y lift to continuous functions on the domain representations? Admissible domain representations A domain representation E of Y is admissible if Y ε E Admissible domain representations A domain representation E of Y is admissible if for each domain D, and each continuous map f : DR → Y where DR is dense in D, Y f ε D E Admissible domain representations A domain representation E of Y is admissible if for each domain D, and each continuous map f : DR → Y where DR is dense in D, Y f ε D E fˉ then f factors through ε. Admissible domain representations are interesting for the following reason: Theorem If D is a dense domain representation of X and E is an admissible domain representation of Y, then every sequentially continuous function f : X → Y lifts to a continuous function f : D → E. f X Y ε δ D E fˉ Not every domain representation is dense What goes wrong if DR is not dense in D? Y f ε D E Not every domain representation is dense What goes wrong if DR is not dense in D? Y f ε D E fˉ If f : X → Y is continuous then f lifts to the closure of DR . Not every domain representation is dense What goes wrong if DR is not dense in D? Y f ε D E fˉ If f : X → Y is continuous then f lifts to the closure of DR . Thus, an alternative is to view f as a partial continuous function from D to E. Partial continuous functions A partial continuous function from D to E is a pair (S, f ) where Partial continuous functions A partial continuous function from D to E is a pair (S, f ) where • S ⊆ D is closed. Partial continuous functions A partial continuous function from D to E is a pair (S, f ) where • S ⊆ D is closed. • f : S → E is continuous. We may now show Theorem E is admissible ⇐⇒ for each domain D and each continuous map f : DR → Y where DR ⊆ D, Y f ε D E We may now show Theorem E is admissible ⇐⇒ for each domain D and each continuous map f : DR → Y where DR ⊆ D, Y f ε D E fˉ f factors through ε via some partial continuous function f . This suggests representing maps from X to Y by partial continuous functions from D to E: This suggests representing maps from X to Y by partial continuous functions from D to E: We say that f : D * E represents f : X → Y if the diagram This suggests representing maps from X to Y by partial continuous functions from D to E: We say that f : D * E represents f : X → Y if the diagram f X ε δ D E fˉ commutes. Y If E is admissible, then Theorem Every sequentially continuous function from X to Y lifts to a partial continuous function from D to E. If E is admissible, then Theorem Every sequentially continuous function from X to Y lifts to a partial continuous function from D to E. If both D and E are admissible then Theorem f : X → Y lifts to a continuous function from D to E if and only if f is sequentially continuous. The domain of partial continuous functions We let • [D * E] = the set of partial continuous functions from D to E The domain of partial continuous functions We let • [D * E] = the set of partial continuous functions from D to E ordered by • f v g ⇐⇒ dom(f ) ⊆ dom(g) and f (x) v g(x) for all x ∈ dom(f ). The domain of partial continuous functions We let • [D * E] = the set of partial continuous functions from D to E ordered by • f v g ⇐⇒ dom(f ) ⊆ dom(g) and f (x) v g(x) for all x ∈ dom(f ). Theorem [D * E] is an domain and [D * E] is effective if D and E are effective. Representing the space of sequentially continuous functions Suppose E is admissible. We let • [X →ω Y] = the space of sequentially continuous functions from X to Y. Representing the space of sequentially continuous functions Suppose E is admissible. We let • [X →ω Y] = the space of sequentially continuous functions from X to Y. To construct a domain representation of [X →ω Y] over [D * E], let Representing the space of sequentially continuous functions Suppose E is admissible. We let • [X →ω Y] = the space of sequentially continuous functions from X to Y. To construct a domain representation of [X →ω Y] over [D * E], let • [D * E]R = {partial continuous functions which represent sequentially continuous maps from X to Y}. Representing the space of sequentially continuous functions Suppose E is admissible. We let • [X →ω Y] = the space of sequentially continuous functions from X to Y. To construct a domain representation of [X →ω Y] over [D * E], let • [D * E]R = {partial continuous functions which represent sequentially continuous maps from X to Y}. We define a map [δ * ε] : [D * E]R → [X →ω Y] by Representing the space of sequentially continuous functions Suppose E is admissible. We let • [X →ω Y] = the space of sequentially continuous functions from X to Y. To construct a domain representation of [X →ω Y] over [D * E], let • [D * E]R = {partial continuous functions which represent sequentially continuous maps from X to Y}. We define a map [δ * ε] : [D * E]R → [X →ω Y] by • [δ * ε](f ) = f ⇐⇒ f represents f . Theorem [D * E] is a domain representation of [X →ω Y]. Theorem [D * E] is a domain representation of [X →ω Y]. Moreover, [D * E] is effective/admissible if D and E are effective/admissible. If we let ADM be the category with objects admissible domain representations X δ D If we let ADM be the category with objects admissible domain representations f X ε δ D E fˉ and morphisms representable maps, then Theorem ADM is Cartesian closed. Y Effectivity and Cartesian closure A partial continuous function f : D * E is effective if we can enumerate the set of compact approximations to f (a) for each compact a ∈ dom(f ). Effectivity and Cartesian closure A partial continuous function f : D * E is effective if we can enumerate the set of compact approximations to f (a) for each compact a ∈ dom(f ). We let EADM be the effective counterpart of ADM. Effectivity and Cartesian closure A partial continuous function f : D * E is effective if we can enumerate the set of compact approximations to f (a) for each compact a ∈ dom(f ). We let EADM be the effective counterpart of ADM. All the constructions on ADM preserve effectivity, except for currying. Effectivity and Cartesian closure A partial continuous function f : D * E is effective if we can enumerate the set of compact approximations to f (a) for each compact a ∈ dom(f ). We let EADM be the effective counterpart of ADM. All the constructions on ADM preserve effectivity, except for currying. The map curry : [D × E * F] * [D * [E * F]] is not effective in general. The partial continuous function curry from [D × E * F] to [D * [E * F]] is effective in many interesting cases: The partial continuous function curry from [D × E * F] to [D * [E * F]] is effective in many interesting cases: Theorem curry is effective if the relation “a ∈ the closure of ER ” is semidecidable for compact a ∈ E. Y ε E ? a∈ Thank you.