1. Three levels of D - completions Zhao Dongsheng Ho Weng Kin, Wee Wen Shih Mathematics and Mathematics Education National Institute of Education, Nayang Technological University, Singapore The 2nd International Conference on Quantitative Logic and Soft Computing 22-25, October, 2010 Jimei University, Xiamen, China

2. Outline: Subcategory and reflectivity Posets, dcpo’s and D – completion of posets Monotone convergence ( d - ) spaces, d-completion of T 0 - spaces Convergence spaces, D-completion

3. 1. Subcategory and reflectivity Given a category C and a subcategory E of C, E is reflective in C if the inclusion functor from E into C has a left adjoint. f Y G(X) ηXηX X C E

4. Examples CRSP KHSP X βXβX Stone–Čech compactification

5. TOP 0 Sober X SX soberification A space Y is sober if for any non-empty irreducible closed set F of Y, there is a unique element a in Y such that F = cl({a})

6. 2. Directed complete posets and dcpo-completion A non-empty subset D of a poset (P, ≤ ) is directed if any two elements in D has an upper bound in D. A poset (P, ≤ ) is directed complete if every directed subset D of P has a supremum (join) sup D. A directed complete poset is also called a dcpo.

7. A mapping f : P → Q between two posets is Scott continuous if it preserves joins of directed sets: f( sup D)=sup f(D) for every directed D for which sup D exists.

8. Let POS ( DCPO) be the category of posets ( dcpo ) and Scott continuous mappings between posets. Question: Is DCPO reflective in POS ?

9. Definition [11] The D-completion of a poset P is a dcpo D(P), together with a Scott continuous mapping η : P → D(P) such that for any Scott continuous f: P → Q into a dcpo Q, there is a unique Scott continuous g: D(P) → Q such that f= g○η

10. A subset U of P is Scott open if the following conditions are satisfied: (i) U=↑U ; (ii) for any directed set D, sup D in U implies D∩U is non- empty The set of all Scott closed (open ) sets of P is denoted by Г(P) ( σ(P)), which is a complete lattice under the inclusion relation.

11. A subset F of a poset P is called D-closed if for any directed subset, whenever sup D exists. The D – closed sets defines a topology--- D - topology. * A subset F is Scott closed iff it is a down set and D-closed

12. Theorem [11] For every poset P, the D - closure of ф={↓x: x is in P} in Г(P), or the smallest subdcpo containing ф, is the D-completion of P. Theorem [11] (1) P is continuous iff D(P) is continuous. (2) P is algebraic iff P is algebraic. (3) Г(P)≡ Г(D(P)).

13. *The directed completion o f continuous posets was considered in [10]. * The dcpo – completion of posets was also considered in [6] and [7] with different motivations. * The local dcpo –completion was first considered in [8] and was revised in [11]

14. For any T 0 space X, the specialization order  on X is defined as : x  y iff x is in cl({y}).

15. 3. d-spaces and d-completion to topological spaces Remark 1.For any poset P, ΣP=(P, σ(P)) is a T 0 space. The specialization order is the original order on P. 2. f: P → Q is Scott continuous iff it is continuous with respect to the Scott topology. So the category POS d is a full subcategory of TOP 0 3.If P is a dcpo, then (i) the specialization of ΣP is directed complete, and (ii) for any directed subset D, D converges to sup D, as a net.

16. Definition A topological space X satisfying (i) and (ii) is called a d - space, or monotone convergence space. Remark: 1.For any poset P, the Scott space ΣP is a d-space iff P is directed complete. 2.Every Sober space is a d-space. 2.If X is a d - space, then every open set U of X is a Scott open set of (X, ≤ ), where is the specialization order on X.

17. poset dcpo P ∑P Q ∑D TOP 0 DTOP 0

18. Question: Is DTOP reflexive in TOP 0 ? Theorem [9] [1] [7] For each T 0 space X there is a d –space D(X) and a continuous mapping η : X → D(X) such that for any continuous f: X → Y into a d-space Y, there is a unique Scott continuous g: D(X) → Y such that f= g∙η.

19. Theorem [7 For any poset, the d-completion D( ∑P) of the space ∑P with the specialization order is the D-completion of P.

20. P ∑P D(∑P) D(P) specialization order Posets Space

21. So the D – completion of topological spaces is a generalization of D – completion of posets

22. 4. Net convergence spaces and D-completion Definition [ 5] A (net ) convergence space is a pair (X, →), where X is a non-empty set and → is a collection of pairs ( S, x ) with S a net in X and x an element in X ( we write S→x if (S, x) belongs to → and say that S converges to x ), such that the following conditions are satisfied: 1.( CONSTANT NET) For any x in X, {x}→x. 2. ( SUBNET) If S → x then S 1 → x for any subnet S 1 of S.

23. * A net S=(x i ) is a subnet of T=(y j ) if for any j 0 there is i 0 such that for any i 0 ≤ i, x i belongs to the j 0 - tail of T. Continuous functions between convergence spaces are defined in the usual way. Let CONV be the category of convergence spaces and continuous functions between them. * CONV is cartesian closed

24. Example Every topological space (X, τ ) defines convergence space (X, ) in the ordinary sense. Such a convergence space is called a topological convergence space. Also the assignment of (X, ) to (X, τ ) extends to a functor from TOP to CONV, setting TOP as a full subcategory of CONV.

25. Definition : A subset U of a convergence space ( X, → ) is open if S=(x i ) → x and x is in U, then x i is in U eventually. The set of all open sets U of ( X, → ) form a topology, called the induced topology and denoted by. The specialization order ≤ of the topological space is called the specialization order of ( X, → ).

26. A space ( X, → ) is up – nice if S=(x i ) converges to a and T=(y j ) is a net satisfying the condition below, then T converges to a: for any i 0 there is j 0 such that for any j 0 ≤ j, y j belongs to ↑{x i : i 0 ≤ i }.

27. A space ( X, → ) is down– nice if S=(x i ) converges to a and b ≤ a, then S converges to b. A space is called nice, if it is both up-nice and down nice. For any net S, let lim S={ x: S → x } be the set of all limit points of S.

28. Definition[5] A convergence space ( X, →) is called a d - space of 1). It is order – nice; 2). For any directed set D={ x i : i in I }, sup D exists and D → sup D ( take D as a net ). 3). For any net, lim S is closed under supremum of directed sets.

29. Example[5] For each poset P, define → by: S=(x i ) → x if there is a directed set E with sup E exist, x ≤ sup E, and for any e in E, e≤ x i eventually. Then (i) (P, ≤ ) is order – nice; (ii) (P, ≤ ) is a d – space iff P is a dcpo; (iii) (P, ≤ ) is topological iff P is a continuous poset.

30. Theorem The category DCONV of d- convergence spaces ( as a subcategory of CONV) is Cartesian closed Example A topological space (X, τ ) is a d- space iff the convergence space (X, ) is a d – space.

31. Remark (1)For any poset P, ( P, → d ) is an IL space if it is topological, iff P is continuous. (2) Every topological convergence space is IL. Definition A space ( X, → ) is called an IL space if it satisfies the Iterated Limit Axiom.

32. Definition[5] A D – completion of a space ( X, → ) is d – space Y together with a continuous map η : X →Y, such that for any continuous f: X → Z from X into a d – space Z, there is a unique continuous h: Y → Z such that f=h ○η.

33. Theorem [5] If X can be embedded into an IL d-space, then X has a d – completion. Corollary Every T 0 topological spaces has a d – completion which is also topological.

34. poset dcpo P ∑P DTOP 0 DCONV TOP 0 X (X, ) CONV

35. Problem: Does every convergence space have a d – completion?

36. References [1] Erchov, Yu. L.: On d-spaces, Theoretical Computer Science, (1999), 224: 59-72. [2] Gierz, G. et al.: Continuous lattices and Domains, Encyclopedia of Mathematics and Its Applications, Vol.93. [3] R. Heckmann: A non- topological view of dcpos as convergence spaces. Theoretical Computer Science, 305(1193), 195 -186. [4] W. Ho, and D. Zhao, Lattices of Scott-closed sets, Comment. Math. Univer. Carolinae, 50(2009), 2:297-314. [5] W. Ho, D. Zhao and W. Wee, d – completion of net convergence space, Proceeding of this conference. [6] A. Jung, M. A. Moshier and S. Vickers, Presenting dcpos and dcpo algebras, (Preprint). [7] Keimel, K. and Lawson, J. D.: D-completions and the d-topology, Annals of Pure and Applied Logic 59(2009), 3:292-306. [8] M. W. Mislove, Local DCPOs, Local CPOs and Local completions, Electronic Notes in Theoretical Computer Science, 20(1999). [9] Wyler, O.: Dedekind complete posets and Scott topologies, In: Lecture Notes In Mathematics, Springer-Verlag, 1981.

37. [10] L. Xu, Continuity of posets via Scott topology and sobrification, Topology Appl., 153(2006), 11:1886-1894. [11] Zhao, D. and Fan, T., dcpo-completion of posets, Theoretical Computer Science, 411(2010), 2167-2173.

38. Thank you !