aa r X i v : . [ m a t h . GN ] N ov A direct approach to K -reflections of T spaces ✩ Xiaoquan Xu a a School of Mathematics and Statistics, Minnan Normal University, Zhangzhou 363000, China
Abstract
In this paper, we provide a direct approach to K -reflections of T spaces. For a full subcategory K of thecategory of all T spaces and a T space X , let K ( X ) = { A ⊆ X : A is closed and for any continuousmapping f : X −→ Y to a K -space Y , there exists a unique y A ∈ Y such that f ( A ) = { y A }} and P H ( K ( X ))the space of K ( X ) endowed with the lower Vietoris topology. It is proved that if P H ( K ( X )) is a K -space,then the pair h X k = P H ( K ( X )) , η X i , where η X : X −→ X k , x
7→ { x } , is the K -reflection of X . We call K an adequate category if for any T space X , P H ( K ( X )) is a K -space. Therefore, if K is adequate, then K is reflective in Top . It is shown that the category of all sober spaces, that of all d -spaces, that of allwell-filtered spaces and the Keimel and Lawson’s category are all adequate, and hence are all reflective in Top . Some major properties of K -spaces and K -reflections of T spaces are investigated. In particular, itis proved that if K is adequate, then the K -reflection preserves finite products of T spaces. Our study alsoleads to a number of problems, whose answering will deepen our understanding of the related spaces andtheir categorical structures. Keywords: K -reflection; K -set; Sober space; Well-filtered space; d -space; Keimel-Lawson category
1. Introduction
Domain theory plays a foundational role in denotational semantics of programming languages. In domaintheory, the d -spaces, well-filtered spaces and sober spaces form three of the most important classes (see [3-17,19-29]). Let Top be the category of all T spaces and Sob the category of all sober spaces. Denote thecategory of all d -spaces and that of all well-filtered spaces respectively by Top d and Top w . It is well-knownthat Sob is reflective in
Top (see [6, 9]). Using d -closures, Wyler [22] proved that Top d is reflective in Top (see also [16, 27]). Later, Ershov [5] showed that the d -completion of X (i.e., the d -reflection of X )can be obtained by adding the closure of directed sets onto X (and then repeating this process by transfiniteinduction). In [16], Keimel and Lawson proved that for a full subcategory K of Top containing Sob , if K has certain properties, then K is reflective in Top . They showed that Top d and some other categorieshave such properties. For quite a long time, it is not known whether Top w is reflective in Top . Recently,following Keimel and Lawson’s method, which originated from Wyler’s method, Wu, Xi, Xu and Zhao [21]gave a positive answer to the above problem. Following Ershov’s method of constructing the d -completionof T spaces, Shen, Xi, Xu and Zhao presented a construction of the well-filtered reflection of T spaces.In this paper, we will provide a direct approach to K -reflections of T spaces. For a full subcategory K of Top containing Sob and a T space X , let K ( X ) = { A ⊆ X : A is closed and for any continuous mapping f : X −→ Y to a K -space Y , there exists a unique y A ∈ Y such that f ( A ) = { y A }} . Endow K ( X ) withthe lower Vietoris topology and denote the resulting space by P H ( K ( X )). We prove that if P H ( K ( X )) is a K -space, then the pair h X k = P H ( K ( X )) , η X i , where η X : X −→ X k , x
7→ { x } , is the K -reflection of X . ✩ This research was supported by the National Natural Science Foundation of China (11661057) and the Natural ScienceFoundation of Jiangxi Province , China (20192ACBL20045)
Email address: [email protected] (Xiaoquan Xu)
Preprint submitted to Topology and its Applications November 27, 2019 e call K an adequate category if for any T space X , P H ( K ( X )) is a K -space. So if K is adequate, then K is reflective in Top . We show that Sob , Top d , Top w and the Keimel and Lawson’s category are alladequate. Therefore, they are all reflective in Top . Some major properties of K -spaces and K -reflections of T spaces are investigated. In particular, it is proved that if K is adequate, then the K -reflection preservesfinite products of T spaces. More precisely, for a finitely family { X i : 1 ≤ i ≤ n } of T spaces, we havethat ( n Q i =1 X i ) k = n Q i =1 X ki (up to homeomorphism). Our study also leads to a number of problems, whoseanswering will deepen our understanding of the related spaces and their categorical structures.
2. Preliminary
In this section, we briefly recall some fundamental concepts and notations that will be used in the paper.Some basic properties of irreducible sets are presented.For a poset P and A ⊆ P , let ↓ A = { x ∈ P : x ≤ a for some a ∈ A } and ↑ A = { x ∈ P : x ≥ a for some a ∈ A } . For x ∈ P , we write ↓ x for ↓{ x } and ↑ x for ↑{ x } . A subset A is called a lower set (resp., an upper set ) if A = ↓ A (resp., A = ↑ A ). Let P ( <ω ) = { F ⊆ P : F is a nonempty finite set } and Fin P = {↑ F : F ∈ P ( <ω ) } .The category of all T spaces is denoted by Top . For X ∈ Top , we use ≤ X to represent the specializa-tion order of X , that is, x ≤ X y iff x ∈ { y } ). In this paper, when a T space X is considered as a poset, theorder always refers to the specialization order if no other explanation. Let O ( X ) (resp., C ( X )) be the set ofall open subsets (resp., closed subsets) of X , and let S u ( X ) = {↑ x : x ∈ X } . Define S c ( X ) = {{ x } : x ∈ X } and D c ( X ) = { D : D ∈ D ( X ) } .A nonempty subset D of a poset P is directed if every two elements in D have an upper bound in D .The set of all directed sets of P is denoted by D ( P ). P is called a directed complete poset , or dcpo for short,if for any D ∈ D ( P ), W D exists in P . A subset U of P is Scott open if (i) U = ↑ U and (ii) for any directedsubset D for which W D exists, W D ∈ U implies D ∩ U = ∅ . All Scott open subsets of P form a topology,and we call this topology the Scott topology on P and denote it by σ ( P ). The space Σ P = ( P, σ ( P )) iscalled the Scott space of P .A T space X is called a d - space (or monotone convergence space ) if X (with the specialization order) isa dcpo and O ( X ) ⊆ σ ( X ) (cf. [6, 22]). Clearly, for a dcpo P , Σ P is a d -space. The category of all d -spaceswith continuous mappings is denoted by Top d .One can directly get the following result (cf. [26]). Proposition 2.1.
For a T space X , the following conditions are equivalent: (1) X is a d -space. (2) D c ( X ) = S c ( X ) . The following result is well-known (cf. [6]).
Lemma 2.2.
Let
P, Q be posets and f : P −→ Q . Then the following two conditions are equivalent: (1) f is Scott continuous, that is, f : Σ P −→ Σ Q is continuous. (2) For any D ∈ D ( P ) for which W D exists, f ( W D ) = W f ( D ) . Lemma 2.3. ([16])
Let f : X −→ Y be a continuous mapping of T spaces. If D ∈ D ( X ) has a supremumto which it converges, then f ( W D ) = W f ( D ) . Corollary 2.4.
Let P be a poset and Y a T space. If f : Σ P −→ Y is continuous, then f : Σ P −→ Σ Y iscontinuous. Let
Poset denote the category of all posets with monotone (i.e. order-preserving) mappings,
DCPO the category of all dcpos with Scott continuous mappings, and
Poset s the category of all posets with Scottcontinuous mappings. Then DCPO is a full subcategory of
Poset s .2 nonempty subset A of a T space X is irreducible if for any { F , F } ⊆ C ( X ), A ⊆ F ∪ F implies A ⊆ F or A ⊆ F . Denote by Irr ( X ) (resp., Irr c ( X )) the set of all irreducible (resp., irreducible closed)subsets of X . Clearly, every subset of X that is directed under ≤ X is irreducible. X is called sober , if forany F ∈ Irr c ( X ), there is a unique point a ∈ X such that F = { a } . The category of all sober spaces withcontinuous mappings is denoted by Sob .The following two lemmas on irreducible sets are well-known.
Lemma 2.5.
Let X be a space and Y a subspace of X . Then the following conditions are equivalent for asubset A ⊆ Y : (1) A is an irreducible subset of Y . (2) A is an irreducible subset of X . (3) cl X A is an irreducible closed subset of X . Lemma 2.6. If f : X −→ Y is continuous and A ∈ Irr ( X ) , then f ( A ) ∈ Irr ( Y ) . Lemma 2.7. ([20])
Let X = Q i ∈ I X i be the product space of T spaces X i ( i ∈ I ) . If A is an irreduciblesubset of X , then cl X ( A ) = Q i ∈ I cl X i ( p i ( A )) , where p i : X −→ X i is the i th projection for each i ∈ I . Lemma 2.8. ([26])
Let X = Q i ∈ I X i be the product space of T spaces X i ( i ∈ I ) and A i ⊆ X i for each i ∈ I . Then the following two conditions are equivalent: (1) Q i ∈ I A i ∈ Irr ( X ) . (2) A i ∈ Irr ( X i ) for each i ∈ I . By Lemma 2.7 and Lemma 2.8, we obtain the following corollary.
Corollary 2.9.
Let X = Q i ∈ I X i be the product space of T spaces X i ( i ∈ I ) . If A ∈ Irr c ( X ) , then A = Q i ∈ I p i ( A ) and p i ( A ) ∈ Irr c ( X i ) for each i ∈ I . For any topological space X , G ⊆ X and A ⊆ X , let G A = { G ∈ G : G T A = ∅} and G A = { G ∈G : G ⊆ A } . The symbols G A and G A will be simply written as A and A respectively if there is noconfusion. The lower Vietoris topology on G is the topology that has { U : U ∈ O ( X ) } as a subbase, andthe resulting space is denoted by P H ( G ). If G ⊆
Irr ( X ), then { G U : U ∈ O ( X ) } is a topology on G . Remark 2.10.
Let X be a T space.(1) If S c ( X ) ⊆ G , then the specialization order on P H ( G ) is the order of set inclusion, and the canonicalmapping η X : X −→ P H ( G ), given by η X ( x ) = { x } , is an order and topological embedding (cf. [6, 9, 19]).(2) The space X s = P H ( Irr c ( X )) with the canonical mapping η X : X −→ X s is the sobrification of X (cf.[6, 9]).(3) P H ( S c ( X )) is a subspace of X s and X is homeomorphic to P S ( S c ( X )) via a homeomorphism x
7→ { x } . Remark 2.11.
Let X be a T space and G ⊆ G ⊆ Irr c ( X ). If G is endowed with the lower Vietoristopology, then the subspace ( G , { U ∩ G : U ∈ O ( X ) } ) is the space P H ( G ), which is a subspace of X s . Inwhat follows, when a subset G in X s (that is G ⊆
Irr c ( X )) is considered as a topological space, the topologyalways refers to the subspace topology of X s if no other explanation.For a space X , a subset A of X is called saturated if A equals the intersection of all open sets containingit (equivalently, A is an upper set in the specialization order). We shall use Q ( X ) to denote the set of allnonempty compact saturated subsets of X and endow it with the Smyth preorder , that is, for K , K ∈ Q ( X ), K ⊑ K iff K ⊆ K . X is called well-filtered if it is T , and for any open set U and filtered family K ⊆ Q ( X ), T K⊆ U implies K ⊆ U for some K ∈K . The category of all well-filtered spaces with continuous mappings isdenoted by Top w . The space P S ( Q ( X )), denoted shortly by P S ( X ), is called the Smyth power space or upperspace of X (cf. [10, 19]). It is easy to see that the specialization order on P S ( X ) is the Smyth order (thatis, ≤ P S ( X ) = ⊑ ). The canonical mapping ξ X : X −→ P S ( X ), x
7→ ↑ x , is an order and topological embedding3cf. [10, 11, 19]). Clearly, P S ( S u ( X )) is a subspace of P S ( X ) and X is homeomorphic to P S ( S u ( X )) via ahomeomorphism x
7→ ↑ x .As in [4], a topological space X is locally hypercompact if for each x ∈ X and each open neighborhood U of x , there is ↑ F ∈ Fin X such that x ∈ int ↑ F ⊆ ↑ F ⊆ U . A space X is called a C - space if for each x ∈ X and each open neighborhood U of x , there is u ∈ X such that x ∈ int ↑ u ⊆ ↑ u ⊆ U . A set K ⊆ X iscalled supercompact if for any arbitrary family { U i : i ∈ I } ⊆ O ( X ), K ⊆ S i ∈ I U i implies K ⊆ U for some i ∈ I . It is easy to check that the supercompact saturated sets of X are exactly the sets ↑ x with x ∈ X (see[11, Fact 2.2]). It is well-known that X is a C -space iff O ( X ) is a completely distributive lattice (cf. [2]). Aspace X is called core compact if O ( X ) is a continuous lattice (cf. [6]). Theorem 2.12. ([17, 26])
Let X be a well-filtered space. Then X is locally compact iff X is core compact.
3. K-sets
For a full subcategory K of Top , the objects of K will be called K -spaces. In [16], Keimel and Lawsonrequired the following properties:(K ) Homeomorphic copies of K -spaces are K -spaces.(K ) All sober spaces are K -spaces or, equivalently, Sob ⊆ K .(K ) In a sober space S, the intersection of any family of K -subspaces is a K -space.(K ) Continuous maps f : S −→ T between sober spaces S and T are K -continuous, that is, for every K -subspace K of T , the inverse image f − ( K ) is a K -subspace of S . Definition 3.1.
A full subcategory K of Top is said to be closed with respect to homeomorphisms if K has (K ). K is called a Keimel-Lawson category if K satisfies (K )-(K ).Clearly, Sob , Top d and Top w are closed with respect to homeomorphisms and satisfy (K ).In what follows, K always refers to a full subcategory Top containing Sob , that is, K has (K ). Fortwo spaces X and Y , we use the symbol X ∼ = Y to represent that X and Y are homeomorphic. Definition 3.2.
A subset A of a T space X is called a K - set , provided for any continuous mapping f : X −→ Y to a K -space Y , there exists a unique y A ∈ Y such that f ( A ) = { y A } . Denote by K ( X ) theset of all closed K -sets of X . X is said to be a K - determined space if Irr c ( X ) = K ( X ) or, equivalently, allirreducible closed sets of X are K -sets (it is easy to check that all K -sets are irreducible, please see Corollary3.4 below).Obviously, a subset A of a space X is a K -set iff A is a K -set. For simplicity, let d ( X ) = Top d ( X ) and WF ( X ) = Top w ( X ). X is called well-filtered determined , WF -determined for short, if all irreducible closedsubsets of X are WF -sets, that is, Irr c ( X ) = WF ( X ). Lemma 3.3.
For a T space X , Sob ( X ) = Irr c ( X ) . Proof . By Lemma 2.6,
Irr c ( X ) ⊆ Sob ( X ). Suppose A ∈ Sob ( X ). Now we show that A is irreducible.Consider the sobrification X s (= P H ( Irr c ( X )) of X and the canonical topological embedding η X : X −→ X s ,given by η X ( x ) = { x } . Then there is a B ∈ Irr c ( X ) such that Irr c ( X ) A = η X ( A ) = { B } = Irr c ( X ) B , andwhence A = B . Thus A ∈ Irr c ( X ). Corollary 3.4.
For a T space X , S c ( X ) ⊆ K ( X ) ⊆ Irr c ( X ) . Proof . Clearly, S c ( X ) ⊆ K ( X ). Since Sob ⊆ K , we have K ( X ) ⊆ Sob ( X ) = Irr c ( X ) by Lemma 3.3.Rudin’s Lemma [18] is a very useful tool in domain theory and non-Hausdorff topology (see [3, 6-9, 11,20, 25, 26]). In [11], Heckman and Keimel presented the following topological variant of Rudin’s Lemma. Lemma 3.5. (Topological Rudin’s Lemma)
Let X be a topological space and A an irreducible subset of theSmyth power space P S ( X ) . Then every closed set C ⊆ X that meets all members of A contains an minimalirreducible closed subset A that still meets all members of A . T space X and K ⊆ Q ( X ), let M ( K ) = { A ∈ C ( X ) : K T A = ∅ for all K ∈ K} (that is, A ⊆ A )and m ( K ) = { A ∈ C ( X ) : A is a minimal menber of M ( K ) } . The following concept was introduced basedon topological Rudin’s Lemma. Definition 3.6. ([20, 26]) Let X be a T space. A nonempty subset A of X is said to have the Rudinproperty , if there exists a filtered family
K ⊆ Q ( X ) such that A ∈ m ( K ) (that is, A is a minimal closed setthat intersects all members of K ). Let RD ( X ) = { A ∈ C ( X ) : A has Rudin property } . The sets in RD ( X )will also be called Rudin sets . Lemma 3.7.
Let X be a T space and Y a well-filtered space. If f : X −→ Y is continuous and A ⊆ X hasRudin property, then there exists a unique y A ∈ X such that f ( A ) = { y A } . Proof . Since A has Rudin property, there exists a filtered family K ⊆ Q ( X ) such that A ∈ m ( K ). Let K f = {↑ f ( K ∩ A ) : K ∈ K} . Then F f ⊆ Q ( Y ) is filtered. For each K ∈ K , since K ∩ A = ∅ , we have ∅ 6 = f ( K ∩ A ) ⊆ ↑ f ( K ∩ A ) ∩ f ( A ). So f ( A ) ∈ M ( K f ). If B is a closed subset of f ( A ) with B ∈ M ( K f ), then B ∩ ↑ f ( K ∩ A ) = ∅ for every K ∈ K . So K ∩ A ∩ f − ( B ) = ∅ for all K ∈ K . It follows that A = A ∩ f − ( B )by the minimality of A , and consequently, f ( A ) ⊆ B . Therefore, f ( A ) = B . Thus f ( A ) ∈ m ( K f ). Since Y is well-filtered, we have T K ∈K ↑ f ( K ∩ A ) ∩ f ( A ) = ∅ . Select a y A ∈ T K ∈K ↑ f ( K ∩ A ) ∩ f ( A ). Then { y A } ⊆ f ( A ) and K ∩ A ∩ f − ( { y A } ) = ∅ for all K ∈ K . It follows that A = A ∩ f − ( { y A } ) by the minimalityof A , and consequently, f ( A ) ⊆ { y A } . Therefore, f ( A ) = { y A } . The uniqueness of y A follows from the T separation of Y . Proposition 3.8.
Let X be a T space. Then S c ( X ) ⊆ D c ( X ) ⊆ RD ( X ) ⊆ WF ( X ) ⊆ Irr c ( X ) . Proof . Obviously, S c ( X ) ⊆ D c . Now we prove that the closure of a directed subset D of X is a Rudin set.Let K D = {↑ d : d ∈ D } . Then K D ⊆ Q ( X ) is filtered and D ∈ M ( K D ). If A ∈ M ( K D ), then d ∈ A forevery d ∈ D , and hence D ⊆ A . So D ∈ m ( K D ). Therefore D ∈ RD ( X ). By Lemma 3.7, RD ( X ) ⊆ WF ( X ).Finally, by Corollary 3.4 (for K = Top w ), we have WF ( X ) ⊆ Irr c ( X ). Example 3.9.
Let X be a countable infinite set and endow X with the cofinite topology (having thecomplements of the finite sets as open sets). The resulting space is denoted by X cof . Then Q ( X cof ) = 2 X \{∅} (that is, all nonempty subsets of X ), and hence X cof is a locally compact and first countable T space. Let K = { X \ F : F ∈ X ( <ω ) } . It is easy to check that K ⊆ Q ( X cof ) is filtered and X ∈ m ( K ). Therefore, X ∈ RD ( X ) but X D c ( X ), and whence RD ( X ) = D c ( X ) and WF ( X ) = D c ( X ). Thus X cof is notwell-filtered (and hence non-sober). Example 3.10.
Let L be the complete lattice constructed by Isbell [15] and K = Top w . Then by [23,Corollary 3.2], Σ L is a well-filtered space, and whence WF ( X ) = S c ( X ). Note that Σ L is not sober.Therefore, by Prpposition 3.8, WF ( X ) = Irr c ( X ) and RD ( X ) = Irr c ( X ). Lemma 3.11.
Let
X, Y be two T spaces. If f : X −→ Y is a continuous mapping and A ∈ K ( X ) , then f ( A ) ∈ K ( Y ) . Proof . Let Z is a K -space and g : Y −→ Z is a continuous mapping. Since g ◦ f : X −→ Z is continuousand A ∈ K ( X ), there is z ∈ Z such that g ( f ( A )) = g ◦ f ( A ) = { z } . Thus f ( A ) ∈ K ( Y ). Lemma 3.12.
Let { X i : 1 ≤ i ≤ n } be a finite family of T spaces and X = n Q i =1 X i the product space. For A ∈ Irr ( X ) , the following conditions are equivalent: (1) A is a K -set. (2) p i ( A ) is a K -set for each ≤ i ≤ n . roof . (1) ⇒ (2): By Lemma 3.11.(2) ⇒ (1): By induction, we need only to prove the implication for the case of n = 2. Let A = cl X p ( A )and A = cl X p ( A ). Then by condition (2), ( A , A ) ∈ K ( X ) × K ( X ). Now we show that the product A × A ∈ K ( X ). Let f : X × X −→ Y a continuous mapping from X × X to a K -space Y . Foreach b ∈ X , X is homeomorphic to X × { b } (as a subspace of X × X ) via the homeomorphism µ b : X −→ X × { b } defined by µ b ( x ) = ( x, b ). Let i b : X × { b } −→ X × X be the embedding of X × { b } in X × X . Then f b = f ◦ i b ◦ µ b : X −→ Y , f b ( x ) = f (( x, b )), is continuous. Since A ∈ K ( X ),there is a unique y b ∈ Y such that f ( A × { b } ) = f b ( A ) = { y b } . Define a mapping g A : X −→ Y by g A ( b ) = y b . For each V ∈ O ( Y ), g − A ( V ) = { b ∈ X : g A ( b ) ∈ V } = { b ∈ X : f b ( A ) ∩ V = ∅} = { b ∈ X : f ( A × { b } ) ∩ V = ∅} = { b ∈ X : f ( A × { b } ) ∩ V = ∅} = { b ∈ X : ( A × { b } ) ∩ f − ( V ) = ∅} . Therefore, for each b ∈ g − A ( V ), there is an a ∈ A such that ( a , b ) ∈ f − ( V ) ∈ O ( X × X ), and hencethere is ( U , U ) ∈ O ( X ) × O ( X ) such that ( a , b ) ∈ U × U ⊆ f − ( V ). It follows that b ∈ U ⊆ g − A ( V ).Thus g A : X −→ Y is continuous. Since A ∈ K ( X ), there is a unique y A ∈ Y such that g A ( A ) = { y A } .Therefore, by Lemma 2.7, we have f (cl X A ) = f ( A × A )= S a ∈ A f ( A × { a } )= S a ∈ A f ( A × { a } )= S a ∈ A { g A ( a ) } = S a ∈ A { g A ( a ) } = g A ( A )= { y A } . Thus cl X A ∈ K ( X ), and hence A is a K -set.By Corollary 2.9 and Lemma 3.12, we get the following result. Corollary 3.13.
Let X = n Q i =1 X i be the product of a finitely family { X i : 1 ≤ i ≤ n } of T spaces. If A ∈ K ( X ) , then A = n Q i =1 p i ( X i ) , and p i ( A ) ∈ K ( X i ) for all ≤ i ≤ n .
4. A direct construction of K -reflections of T spaces In Section 4, we give a direct construction of the K -reflections of T spaces and investigate some basicproperties of K -spaces and K -reflections. In particular, it is proved that if K ia an adequate category, thenthe K -reflection preserves finite products of T spaces. Definition 4.1.
Let X be a T space. A K - reflection of X is a pair h e X, µ i consisting of a K -space e X anda continuous mapping µ : X −→ e X satisfying that for any continuous mapping f : X −→ Y to a K -space,there exists a unique continuous mapping f ∗ : e X −→ Y such that f ∗ ◦ µ = f , that is, the following diagram6ommutes. X f (cid:31) (cid:31) ❅❅❅❅❅❅❅❅ µ / / e X f ∗ (cid:15) (cid:15) Y By a standard argument, K -reflections, if they exist, are unique up to homeomorphism. We shall use X k to denote the space of the K -reflection of X if it exists.By Corollary 3.4, { K ( X ) U : U ∈ O ( X ) } is a topology on K ( X ). In the following, let η kX : X −→ P H ( K ( X )), η kX ( x ) = { x } , be the canonical mapping from X to P H ( K ( X )). Lemma 4.2.
The canonical mapping η kX : X −→ P H ( K ( X )) is a topological embedding. Proof . For U ∈ O ( X ), we have( η kX ) − ( U ) = { x ∈ X : ↓ x ∈ U } = { x ∈ X : x ∈ U } = U, so η kX is continuous. In addition, we have η kX ( U ) = {↓ x : x ∈ U } = {↓ x : ↓ x ∈ U } = U ∩ η kX ( X ) , which implies that η kX is an open mapping to η kX ( X ), as a subspace of P H ( K ( X )). As η kX is injective, η kX isa topological embedding. Lemma 4.3.
For a T space X be and A ⊆ X , η kX ( A ) = η kX (cid:0) A (cid:1) = A = A in P H ( K ( X )) . Proof . Clearly, η kX ( A ) ⊆ A ⊆ A , η kX (cid:0) A (cid:1) ⊆ A and A is closed in P H ( K ( X )). It follows that η kX ( A ) ⊆ A ⊆ A and η kX ( A ) ⊆ η kX (cid:0) A (cid:1) ⊆ A. To complete the proof, we need to show A ⊆ η kX ( A ). Let F ∈ A . Suppose U ∈ O ( X ) such that F ∈ U (note that F ∈ K ( X )), that is, F ∩ U = ∅ . Since F ⊆ A , we have A ∩ U = ∅ . Let a ∈ A ∩ U . Then ↓ a ∈ U ∩ η kX ( A ) = ∅ . This implies that F ∈ η kX ( A ). Thus A ⊆ η kX ( A ). Lemma 4.4.
Let X be a T space and A a nonempty subset of X . Then the following conditions areequivalent: (1) A is irreducible in X . (2) A is irreducible in P H ( K ( X )) . (3) A is irreducible in P H ( K ( X )) . Proof . (1) ⇒ (3): Assume A is irreducible. Then η kX ( A ) is irreducible in P H ( K ( X )) by Lemma 2.6 andLemma 4.2. By Lemma 2.5 and Lemma 4.3, A = η kX ( A ) is irreducible in P H ( K ( X )).(3) ⇒ (1): Assume A is irreducible. Let A ⊆ B ∪ C with B, C ∈ C ( X ). By Corollary 3.4, K ( X ) ⊆ Irr c ( X ), and consequently, we have A ⊆ B ∪ C . Since A is irreducible, A ⊆ B or A ⊆ C , showingthat A ⊆ B or A ⊆ C , and consequently, A ⊆ B or A ⊆ C , proving A is irreducible.(2) ⇔ (3): By Lemma 2.5 and Lemma 4.3.For the K -reflections of T spaces, the following lemma is crucial.7 emma 4.5. Let X be a T space and f : X −→ Y a continuous mapping from X to a well-filtered space Y . Then there exists a unique continuous mapping f ∗ : P H ( K ( X )) −→ Y such that f ∗ ◦ η kX = f , that is,the following diagram commutes. X f $ $ ❏❏❏❏❏❏❏❏❏❏❏ η kX / / P H ( K ( X )) f ∗ (cid:15) (cid:15) Y Proof . For each A ∈ K ( X ), there exists a unique y A ∈ Y such that f ( A ) = { y A } . Then we can define amapping f ∗ : P H ( K ( X )) −→ Y by ∀ A ∈ K ( X ) , f ∗ ( A ) = y A . Claim 1: f ∗ ◦ η kX = f .Let x ∈ X . Since f is continuous, we have f (cid:16) { x } (cid:17) = f ( { x } ) = { f ( x ) } , so f ∗ (cid:16) { x } (cid:17) = f ( x ). Thus f ∗ ◦ η kX = f .Claim 2: f ∗ is continuous.Let V ∈ O ( Y ). Then ( f ∗ ) − ( V ) = { A ∈ K ( X ) : f ∗ ( A ) ∈ V } = { A ∈ K ( X ) : { f ∗ ( A ) } ∩ V = ∅} = { A ∈ K ( X ) : f ( A ) ∩ V = ∅} = { A ∈ K ( X ) : f ( A ) ∩ V = ∅} = { A ∈ K ( X ) : A ∩ f − ( V ) = ∅} = f − ( V ) , which shows that ( f ∗ ) − ( V ) is open in P H ( K ( X )). Thus f ∗ is continuous.Claim 3: The mapping f ∗ is unique such that f ∗ ◦ η kX = f .Assume g : P H ( K ( X )) −→ Y is a continuous mapping such that g ◦ η kX = f . Let A ∈ K ( X ). We needto show g ( A ) = f ∗ ( A ). Let a ∈ A . Then { a } ⊆ A , implying that g ( { a } ) ≤ Y g ( A ), that is, g (cid:16) { a } (cid:17) = f ( a ) ∈ { g ( A ) } . Thus { f ∗ ( A ) } = f ( A ) ⊆ { g ( A ) } . In addition, since A ∈ η kX ( A ) and g is continuous, g ( A ) ∈ g (cid:16) η kX ( A ) (cid:17) ⊆ g ( η kX ( A )) = f ( A ) = { f ∗ ( A ) } , which implies that { g ( A ) } ⊆ { f ∗ ( A ) } . So { g ( A ) } = { f ∗ ( A ) } .Since Y is T , g ( A ) = f ∗ ( A ). Thus g = f ∗ .From Lemma 4.5 we deduce the following main result of this paper. Theorem 4.6.
Let X be a T space. If P H ( K ( X )) is a K -space, then the pair h X k = P H ( K ( X )) , η kX i ,where η kX : X −→ X k , x
7→ { x } , is the K -reflection of X . Definition 4.7. K is called adequate if for any T space X , P H ( K ( X )) is a K -space. Corollary 4.8. If K is adequate, then K is reflective in Top . Corollary 4.9. If K is adequate, then for any T spaces X, Y and any continuous mapping f : X −→ Y ,there exists a unique continuous mapping f k : X k −→ Y k such that f k ◦ η kX = η kY ◦ f , that is, the followingdiagram commutes. X f (cid:15) (cid:15) η kX / / X kf k (cid:15) (cid:15) Y η kY / / Y k For each A ∈ K ( X ) , f k ( A ) = f ( A ) . K : Top −→ K , which is the left adjoint to the inclusion functor I : K −→ Top . Corollary 4.10.
Suppose that K is adequate and closed with respect to homeomorphisms. Then for any T space X , the following conditions are equivalent: (1) X is a K -space. (2) K ( X ) = S c ( X ) , that is, for each A ∈ K ( X ) , there exists an x ∈ X such that A = { x } . (3) X ∼ = X k . Proof . (1) ⇒ (2): Considering the identity id X : X −→ X .(2) ⇒ (3): X k = P H ( K ( X )) = P H ( S c ( X )) ∼ = X via a homeomorphism x
7→ { x } .(3) ⇒ (1): By the adequateness of K , X k = P H ( K ( X )) is a K -space. Since K is closed with respect tohomeomorphisms and X ∼ = X k , X is a K -space. Corollary 4.11.
Let K be adequate and closed with respect to homeomorphisms. Then a retract of a K -spaceis a K -space. Proof . Suppose that Y is a retract of a K -space X . Then there are continuous mappings f : X −→ Y and g : Y −→ X such that f ◦ g = id Y . Let B ∈ K ( Y ), then by Lemma 3.11 and Corollary 4.10, there existsa unique x B ∈ X such that g ( B ) = { x B } . It follows that B = f ◦ g ( B ) = f ( g ( B )) = f ( { x B } ) = { f ( x B ) } .Therefore, K ( Y ) = S c ( X ), and hence Y is a K -space by Corollary 4.10. Theorem 4.12.
For an adequate K and a finitely family { X i : 1 ≤ i ≤ n } of T spaces, ( n Q i =1 X i ) k = n Q i =1 X ki ( up to homeomorphism ) . Proof . Let X = n Q i =1 X i . By Corollary 3.13, we can define a mapping γ : P H ( K ( X )) −→ n Q i =1 P H ( K ( X i )) by ∀ A ∈ K ( X ), γ ( A ) = ( p ( A ) , p ( A ) , ..., p n ( A )).By Lemma 3.12 and Corollary 3.13, γ is bijective. Now we show that γ is a homeomorphism. For any( U , U , ..., U n ) ∈ O ( X ) × O ( X ) × ... × O ( X n ), by Lemma 3.12 and Corollary 3.13, we have γ − ( U × U × ... × U n ) = { A ∈ K ( X ) : γ ( A ) ∈ U × U × ... × U n } = { A ∈ K ( X ) : p ( A ) ∩ U = ∅ , p ( A ) ∩ U = ∅ , ..., p n ( A ) ∩ U n = ∅} = { A ∈ K ( X ) : A ∩ U × U × ... × U n = ∅} = U × U × ... × U n ∈ O ( P H ( K ( X )) , and γ ( U × U × ... × U n ) = { γ ( A ) : A ∈ K ( X ) and A ∩ U × U × ... × U n = ∅} = { γ ( A ) : A ∈ K ( X ) , and p ( A ) ∩ U = ∅ , p ( A ) ∩ U = ∅ , ..., p n ( A ) ∩ U n = ∅} = U × U × ... × U n ∈ O ( n Q i =1 P H ( K ( X i ))) . Therefore, γ : P H ( K ( X )) −→ n Q i =1 P H ( K ( X i )) is a homeomorphism, and hence X k (= P H ( K ( X )) and n Q i =1 X ki (= n Q i =1 P H ( K ( X i )) are homeomorphic. Theorem 4.13.
Suppose that K is adequate and closed with respect to homeomorphisms. Then for anyfamily { X i : i ∈ I } of T spaces, the following two conditions are equivalent: (1) The product space Q i ∈ I X i is a K -space. (2) For each i ∈ I , X i is a K -space. roof . (1) ⇒ (2): For each i ∈ I , X i is a retract of Q i ∈ I X i . By Corollary 4.11, X i is a K -space.(2) ⇒ (1): Let X = Q i ∈ I X i . Suppose A ∈ K ( X ). Then by Corollary 2.9, Corollary 3.4 and Lemma3.11, A ∈ Irr c ( X ) and for each i ∈ I , p i ( A ) ∈ K ( X i ), and consequently, there is a u i ∈ X i such that p i ( A ) = cl X i { u i } by condition (2) and Corollary 4.10. Let u = ( u i ) i ∈ I . Then by Corollary 2.9 and [1,Proposition 2.3.3]), we have A = Q i ∈ I p i ( A ) = Q i ∈ I cl u i { u i } = cl X { u } . Thus K ( X ) = S c ( X ). It followsthat X is a K -space by Corollary 4.10. Theorem 4.14.
For an adequate K and a T space X , the following conditions are equivalent: (1) X k is the sobrification of X , in other words, the K -reflection of X and sobrification of X are the same. (2) X k is sober. (3) K ( X ) = Irr c ( X ) . Proof . (1) ⇒ (2): Trivial.(2) ⇒ (3): By Corollary 3.4, K ( X ) ⊆ Irr c ( X ). Now we show that Irr c ( X ) ⊆ K ( X ). Let η kX : X −→ X k be the canonical topological embedding defined by η kX ( x ) = { x } (see Theorem 4.6). Since the pair h X s , η sX i ,where η sX : X −→ X s = P H ( Irr c ( X )), x
7→ { x } , is the soberification of X and X k is sober, there exists aunique continuous mapping ( η kX ) ∗ : X s −→ X k such that ( η kX ) ∗ ◦ η sX = η kX , that is, the following diagramcommutes. X η kX ❇❇❇❇❇❇❇ η sX / / X s ( η kX ) ∗ (cid:15) (cid:15) X k So for each A ∈ Irr c ( X ), there exists a unique B ∈ K ( X ) such that ↓ K ( X ) A = η kX ( A ) = { B } = ↓ K ( X ) B .Clearly, we have B ⊆ A . On the other hand, for each a ∈ A, { a } ∈↓ K ( X ) A = ↓ K ( X ) B , and whence { a } ⊆ B .Thus A ⊆ B , and consequently, A = B . Thus A ∈ K ( X ).(3) ⇒ (1): If K ( X ) = Irr c ( X ), then X k = P H ( K ( X )) = P H ( Irr c ( X )) = X s , with η kX = η sX : X −→ X k ,is the sobrification of X . Proposition 4.15.
For an adequate K and a T space X , X is compact iff X k is compact. Proof . By Corollary 3.4, we have S c ( X ) ⊆ K ( X ) ⊆ Irr c ( X ). Suppose that X is compact. For { U i : i ∈ I } ⊆ O ( X ), if K ( X ) ⊆ S i ∈ I U i , then X ⊆ S i ∈ I U i since S c ( X ) ⊆ K ( X ), and consequently, X ⊆ S i ∈ I U i for some I ∈ I ( <ω ) . It follows that K ( X ) ⊆ S i ∈ I U i . Thus X k is compact. Conversely, if X k is compactand { V j : j ∈ J } is a open cover of X , then K ( X ) ⊆ S j ∈ J V j . By the compactness of X k , there is a finitesubset J ⊆ J such that K ( X ) ⊆ S j ∈ J V j , and whence X ⊆ S j ∈ J V j , proving the compactness of X .For an adequate K , since S c ( X ) ⊆ K ( X ) ⊆ Irr c ( X ) (see Corollary 3.4), the correspondence U ↔ U (= K ( X ) U ) is a lattice isomorphism between O ( X ) and O ( X k ). Therefore, we have the following result. Proposition 4.16.
Let K be adequate and X a T space. Then (1) X is locally hypercompact iff X k is locally hypercompact. (2) X is a C-space iff X k is a C-space. (3) X is core compact iff X k is core compact. Remark 4.17. If K is adequate and K ⊆ Top w , then for a T space X , by Theorem 2.12 and Proposition4.16, the following conditions are equivalent:(1) X is core compact.(2) X k is core compact. 103) X k is locally compact. Remark 4.18.
In [14] (see also [6, Exercise V-5.25]) Hofmann and Lawson given a core compact T space X but not locally compact. By Remark 4.17 and Theorem 5.14, X s and X w are locally compact. So thelocal compactness of X w (or X s ) does not imply the local compactness of X . Definition 4.19. K is said to be a
Smyth category , if for any K -space X , the Smyth power space P S ( X )is a K -space. Proposition 4.20. Sob and
Top w are Smyth categories. Proof . By [11, Theorem 3.13],
Sob is a Smyth category.
Top w is a Smyth category by [25, Theorem 3] or[26, Theorem 5.3]. Remark 4.21.
Let X be any d -space but not well-filtered (see Example 3.9). Then by [25, Theorem 5], P S ( X ) is not a d -space. So Top d is not a Smyth category. Theorem 4.22.
Let K be an adequate Smyth category. For a T space X , if P S ( X ) is K -determined, then X is K -determined. Proof . Let A ∈ Irr c ( X ), Y a K -space and f : X −→ Y a continuous mapping. Then ξ X ( A ) = A ∈ Irr c ( P S ( X )) by Lemma 2.5 and Lemma 2.6, and hence A ∈ K ( P S ( X )) since P S ( X ) is K -determined,where ξ X : X −→ P S ( X ), x
7→ ↑ x . Define a mapping P S ( f ) : P S ( X ) −→ P S ( Y ) by ∀ K ∈ Q ( X ) , P S ( f )( K ) = ↑ f ( K ) . Claim 1: P S ( f ) ◦ ξ X = ξ Y ◦ f .For each x ∈ X , we have P S ( f ) ◦ ξ X ( x ) = P S ( f )( ↑ x ) = ↑ f ( x ) = ξ Y ◦ f ( x ) , that is, the following diagram commutes. X f (cid:15) (cid:15) ξ X / / P S ( X ) P S ( f ) (cid:15) (cid:15) Y ξ Y / / P S ( Y )Claim 2: P S ( f ) : P S ( X ) −→ P S ( Y ) is continuous.Let V ∈ O ( Y ). We have P S ( f ) − ( V ) = { K ∈ Q ( X ) : P S ( f )( K ) = ↑ f ( K ) ⊆ V } = { K ∈ Q ( X ) : K ⊆ f − ( V ) } = f − ( V ) , which is open in P S ( X ). This implies that P S ( f ) is continuous.Since K is a Smyth category, P S ( Y ) is a K -space. By the continuity of P S ( f ) and A ∈ K ( P S ( X )),there exists a unique Q ∈ Q ( Y ) such that P S ( f )( A ) = { Q } .Claim 3: Q is supercompact.Let { U j : j ∈ J } ⊆ O ( X ) with Q ⊆ S j ∈ J U j , i.e., Q ∈ S j ∈ J U j . Note that P S ( f )( A ) = {↑ f ( a ) : a ∈ A } , thus {↑ f ( a ) : a ∈ A } ∩ S j ∈ J U j = ∅ . Then there exists a ∈ A and j ∈ J suchthat Q ⊆ ↑ f ( a ) ⊆ U j .Hence, by [11, Fact 2.2], there exists y Q ∈ Y such that Q = ↑ y Q .Claim 4: f ( A ) = { y Q } .Note that {↑ f ( a ) : a ∈ A } = {↑ y Q } . Thus for each y ∈ f ( A ), ↑ y ∈ {↑ y Q } , showing that ↑ y Q ⊆ ↑ y ,i.e., y ∈ { y Q } . This implies that f ( A ) ⊆ { y Q } . In addition, since ↑ y Q ∈ {↑ f ( a ) : a ∈ A } = f ( A ), ↑ y Q ∩ f ( A ) = ∅ . This implies that y Q ∈ f ( A )m, and whence f ( A ) = { y Q } . Thus A ∈ K ( X ). Therefore, byCorollary 3.4, K ( X ) = Irr c ( X ), proving that X is K -determined.11 . Applications This section is devoted to giving some applications of the results of Section 4 to
Sob , Top d , Top w andthe Keimel-Lawson category.First, we consider the case of K = Sob . For a T space X , by Lemma 3.3, Sob ( X ) = Irr c ( X ). It iswell-known that P H ( Irr c ( X )) is sober (see, e.g., [6, 9]). In fact, for any A ∈
Irr c ( P H ( Irr c ( X ))), let A = ∪A .Then A ∈ Irr c ( X ) and A = { A } in P H ( Irr c ( X )). Thus P H ( Irr c ( X )) is sober. Therefore, by Proposition 4.20,we get the following well-known result. Proposition 5.1. Sob is an adequate Smyth category. Therefore, for any T space X , X s = P H ( Irr c ( X )) with the canonical mapping η X : X −→ X s is the sobrification of X . It follows from Proposition 5.1 that
Sob is reflective in
Top (cf. [6]). Proposition 5.2. ([12, 13])
For a family { X i : i ∈ I } of T spaces, ( Q i ∈ I X i ) s = Q i ∈ I X si ( up to homeo-morphism ) . Proof . Let X = Q i ∈ I X i . By Lemma 2.8 and Corollary 2.9, we can define a bijective mapping β : P H ( Irr c ( X )) −→ Q i ∈ I P H ( Irr c ( X i )) by ∀ A ∈ Irr c ( X ), β ( A ) = ( p i ( A )) i ∈ I .Now we show that β is a homeomorphism. Let q i : Q i ∈ I P H ( Irr c ( X i )) −→ P H ( Irr c ( X i )) be the i thprojection ( i ∈ I ). For any J ∈ I ( <ω ) and ( U i ) i ∈ J ∈ Q i ∈ J O ( X i ), by Lemma 2.8 and Corollary 2.9, we have β − ( T i ∈ J q − i ( U i )) = { A ∈ Irr c ( X ) : β ( A ) ∈ T i ∈ J q − i ( U i ) } = { A ∈ Irr c ( X ) : p i ( A ) ∩ U i = ∅ for each i ∈ J } = { A ∈ Irr c ( X ) : A ∩ T i ∈ J p − i ( U i ) = ∅} = T i ∈ J p − i ( U i ) ∈ O ( P H ( Irr c ( X )) , and β ( T i ∈ J p − i ( U i )) = { β ( A ) : A ∈ Irr c ( X ) and A ∩ T i ∈ J p − i ( U i ) = ∅} = T i ∈ J p − i ( U i ) ∈ O ( Q i ∈ I P H ( Irr c ( X i ))) . Therefore, β : P H ( Irr c ( X )) −→ Q i ∈ I P H ( Irr c ( X i )) is a homeomorphism, and hence X s (= P H ( Irr c ( X ))and Q i ∈ I X si (= Q i ∈ I P H ( Irr c ( X i )) are homeomorphic.By Theorem 4.13 and Proposition 5.1, we get the following well-known result (see, e.g. [6, ExerciseO-5.16]). Corollary 5.3.
For a family { X i : i ∈ I } of T spaces, the following two conditions are equivalent: (1) The product space Q i ∈ I X i is sober. (2) For each i ∈ I , X i is sober. Second, we consider the case of K = Top d . Theorem 5.4. Top d is adequate. Therefore, for any T space X , X d = P H ( d ( X )) with the canonicalmapping η X : X −→ X d is the d -reflection of X . Proof . Suppose that X be a T space. We show that P H ( d ( X )) is a d -space. Since X is T , one can directlydeduce that P H ( d ( X )) is T . Let { A d : d ∈ D } ⊆ d ( X ) be a directed family. Let A = S d ∈ D A d . We checkthat A ∈ d ( X ). For any continuous mapping f : X −→ Y to a d -space Y and d ∈ D , by A d ∈ d ( X ), there12s a y d ∈ Y such that f ( A d ) = { y d } . Since { A d : d ∈ D } ⊆ d ( X ) is directed, { y d : d ∈ D } ∈ D ( Y ). ByProposition 2.1, there is a y ∈ Y such that { y d : d } = { y } . Therefore, we have f ( A ) = f ( S d ∈ D A d )= f ( S d ∈ D A d )= S d ∈ D f ( A d )= S d ∈ D f ( A d )= S d ∈ D { y d } = { y d : d ∈ D } = { y } . Thus A ∈ d ( X ). Clearly, { A d : d ∈ D } = { A } in P H ( d ( X )). By Proposition 2.1 again, P H ( d ( X )) is a d -space.By Lemma 4.5, the pair h X d = P H ( d ( X )) , η dX i , where η dX : X −→ X d , x
7→ { x } , is the d -reflection of X . Corollary 5.5. ([5, 16, 22])
Top d is reflective in Top . From Theorem 4.13 and Theorem 5.4 we deduce the following known result.
Corollary 5.6.
For a family { X i : i ∈ I } of T spaces, the following two conditions are equivalent: (1) The product space Q i ∈ I X i is a d -space. (2) For each i ∈ I , X i is a d -space. By Theorem 5.2 and Theorem 5.4, we get the following two results, which were proved by Keimel andLawson using d -closures in [16]. Corollary 5.7. ([16])
For a finitely family { X i : 1 ≤ i ≤ n } of T spaces, ( n Q i =1 X i ) d = n Q i =1 X di ( up tohomeomorphism ) . Corollary 5.8. ([16])
For a T space X , if ξ X : X −→ Σ d ( X ) , ξ X ( x ) = { x } , is continuous, then the d -reflection X d of X is the Scott space Σ d ( X ) . Proof . By Theorem 5.4, X d = P H ( d ( X )) with η X : X −→ X d is the d -reflection of X , and consequently, d ( X ) (with respect to the specialization order or, equivalently, the order of set inclusion) is a dcpo, Σ d ( X )is a d -space, and O ( X d ) ⊆ σ ( d ( X )). Since ξ X : X −→ Σ d ( X ) is continuous, there is a unique continuousmapping ( ξ X ) d : P H ( d ( X )) −→ Σ d ( X ) such that ( ξ X ) d ◦ η X = ξ X , that is, the following diagram commutes. X ξ X $ $ ❍❍❍❍❍❍❍❍❍❍ η X / / P H ( d ( X )) ( ξ X ) d (cid:15) (cid:15) Σ d ( X )For each A ∈ d ( X ), by Lemma 4.3 and the proof of Lemma 4.5, there exists a unique B ∈ d ( X ) suchthat ↓ d ( X ) A = A = ξ X ( A ) = { B } = ↓ d ( X ) B . Therefore, A = B (note that S c ( X ) ⊆ d ( X )), and hence( ξ X ) d ( A ) = A . It follows that σ ( d ( X )) ⊆ O ( X d ). Thus O ( X d ) = σ ( d ( X )).The following result shows that the d -reflection space of Scott space of a poset P is the Scott spaceΣ d (Σ P )) of the dcpo d (Σ P ). Corollary 5.9.
For any poset P , d (Σ P ) is a dcpo and the d -reflection (Σ P ) d of Σ P is the Scott space Σ d (Σ P )) with the canonical mapping η P : Σ P −→ Σ d ( X ) , given by η P ( x ) = cl σ ( P ) { x } for each x ∈ X . roof . By Corollary 2.4, Theorem 5.4 and Corollary 5.9.
Corollary 5.10. ([28])
DCPO is reflective in
Poset s . Definition 5.11. ([28]) A
DCPO - completion of a poset P , D -completion of P for short, is a pair h e P , η i consisting of a dcpo e P and a Scott continuous mapping η : P −→ e P , such that for any Scott continuousmapping f : P −→ Q to a dcpo Q , there exists a unique Scott continuous mapping e f : e P −→ Q such that e f ◦ η = f , that is, the following diagram commutes. P f (cid:30) (cid:30) ❃❃❃❃❃❃❃❃ η / / e P e f (cid:15) (cid:15) Q D -completions, if they exist, are unique up to isomorphism. We shall use D ( P ) to denote the D -completion of P if it exists.In [28], using the D -topologies defined in [28], Zhao and Fan proved that for any poset P , the D -completion of P exists. As Keimel and Lawson pointed out in [16] that the D -completion of a poset P is aspecial case of the d -reflection of a certain T space. More precisely, the d -reflection of Scott space Σ P . Proposition 5.12.
For a poset P , D ( P ) = d (Σ P ) with the canonical mapping η P : P −→ D ( P ) , η P ( x ) = cl σ ( P ) { x } , is the D -completion of P . Proof . By Theorem 5.4, (Σ P ) d = P H ( d (Σ P )) with the canonical mapping η P : Σ P −→ (Σ P ) d , η P ( x ) = cl σ ( P ) { x } , is the d -reflection of Σ P . By Lemma 2.2 and Corollary 5.9, D ( P ) = d (Σ P ) with the canonicalmapping η P : P −→ D ( P ) is the D -completion of P .Now we consider the case of K = Top w . Lemma 5.13.
Let X be a T space and C ∈ C ( X ) . Then the following conditions are equivalent: (1) C ∈ K ( X ) . (2) C ∈ K ( P H ( K ( X ))) . Proof . (1) ⇒ (2): By Propositions 3.11, Lemma 4.2 and Lemma 4.3.(2) ⇒ (1). Let Y be a K -space and f : X −→ Y a continuous mapping. By Lemma 4.5, there exists acontinuous mapping f ∗ : P H ( K ( X )) −→ Y such that f ∗ ◦ η X = f . Since C = η X ( C ) is a K -set and f ∗ iscontinuous, there exists a unique y C ∈ Y such that f ∗ (cid:16) η X ( C ) (cid:17) = { y C } . Furthermore, we have { y C } = f ∗ (cid:16) η X ( C ) (cid:17) = f ∗ ( η X ( C )) = f ( C ) . So C is a K -set. Theorem 5.14. Top w is adequate. Therefore, for any T space X , X w = P H ( WF ( X )) with the canonicalmapping η X : X −→ X w is the well-filtered reflection of X . Proof . Suppose that X be a T space. We show that P H ( WF ( X )) is well-filtered. Since X is T , one candirectly check that P H ( WF ( X )) is T . Let {K i : i ∈ I } ⊆ Q ( P H ( WF ( X ))) be a filtered family and U ∈ O ( X )such that T i ∈ I K i ⊆ U . We need to show K i ⊆ U for some i ∈ I . Assume, on the contrary, K i * U ,i.e., K i ∩ ( X \ U ) = ∅ , for any i ∈ I .Let A = { C ∈ C ( X ) : C ⊆ X \ U and K i ∩ C = ∅ for all i ∈ I } . Then we have the following two facts.(a1) A 6 = ∅ because X \ U ∈ A .(a2) For any filtered family F ⊆ A , T F ∈ A . 14et F = T F . Then F ∈ C ( X ) and F ⊆ X \ U . Assume, on the contrary, F / ∈ A . Then there exists i ∈ I such that K i ∩ F = ∅ . Note that F = T C ∈F C , implying that K i ⊆ S C ∈F ( X \ C ) and { ( X \ C ) : C ∈ F} is a directed family since F is filtered. Then there is C ∈ F such that K i ⊆ ( X \ C ),i.e., K I ∩ C = ∅ , contradicting C ∈ A . Hence F ∈ A .By Zorn’s Lemma, there exists a minimal element C m in A such that C m intersects all members of K . Clearly, C m is also a minimal closure set that intersects all members of K , hence is a Rudin set in P H ( K ( X )). By Proposition 3.8 and Lemma 5.13, C m ∈ WF ( X ). So C m ∈ C m ∩ T K 6 = ∅ . It follows that T K * ( X \ C m ) ⊇ U , which implies that T K * U , a contradiction.By Lemma 4.5, the pair h X w = P H ( WF ( X )) , η wX i , where η wX : X −→ X w , x
7→ { x } , is the well-filteredreflection of X . Corollary 5.15. ([20, 21, 26])
Top w is reflective in Top . By Theorem 4.13 and Theorem 5.14, we have the following result.
Corollary 5.16. ([20, 26])
For a family { X i : i ∈ I } of T spaces, the following two conditions are equivalent: (1) The product space Q i ∈ I X i is well-filtered. (2) For each i ∈ I , X i is a well-filtered. Finally, we consider the case that K is a Keimel-Lawson category. Theorem 5.17.
Let K be a Keimel-Lawson category. Then K is adequate. Therefore, for any T space X , X k = P H ( K ( X )) with the canonical mapping η kX : X −→ X k is the K -reflection of X . Proof . For any K -space Y and continuous mapping f : X −→ Y , let j kX : X k −→ X s be the inclusionmapping (note that K ( X ) ⊆ Irr c ( X )). By Lemma 4.5 and Proposition 5.1, there is a unique continuousmappings f k : X k −→ Y such that f k ◦ η kX = f , and a unique continuous mapping f ∗ : X s −→ Y s suchthat f ∗ ◦ kX = η Y ◦ f k , that is, the following diagram commutes. X f (cid:15) (cid:15) η kX / / X kf k (cid:15) (cid:15) j kX / / X sf ∗ (cid:15) (cid:15) Y id Y / / Y η sY / / Y s For each A ∈ K ( X ), { f k ( A ) } = { f ∗ ( A ) } = f ( A ). Let [ X → K ] = { f : X −→ Y | Y is a K -spaceand f is continuous } . We have that K ( X ) = T f ∈ [ X → K ] ( f ∗ ) − ( {{ y } : y ∈ Y } ), and whence P H ( K ( X )) = T f ∈ [ X → K ] P H (( f ∗ ) − ( P H ( S c ( Y )))) (see Remark 2.11). P H ( S c ( Y )) is homeomorphic to Y , and hence it is a K -space by (K ). For each f ∈ [ X → K ], by (K ) and (K ), P H (( f ∗ ) − ( P H ( S c ( Y )))) (as a subspace of X s ) is a K -space. Finally, by (K ), P H ( K ( X )) is a K -space. Thus K is adequate. By Lemma 4.5, the pair h X k = P H ( K ( X )) , η wX i , where η kX : X −→ X k , x
7→ { x } , is the K -reflection of X . Corollary 5.18. ([16])
Every Keimel-Lawson category K is reflective in Top . From Theorem 5.2 and Theorem 5.17 we deduce the following corollary.
Corollary 5.19.
Let K be a Keimel-Lawson category. For a finitely family { X i : 1 ≤ i ≤ n } of T spaces, ( n Q i =1 X i ) k = n Q i =1 X ki ( up to homeomorphism ) . By Theorem 4.13 and Theorem 5.17, we get the following result.
Corollary 5.20.
Let K be a Keimel-Lawson category. Then for a family { X i : i ∈ I } of T spaces, thefollowing two conditions are equivalent: (1) The product space Q i ∈ I X i is a K -space. (2) For each i ∈ I , X i is a K -space. . Conclusion In this paper, we provided a direct approach to K -reflections of T spaces. For a full subcategory K of Top containing Sob and a T space X , it was proved that if P H ( K ( X )) is a K -space, then the pair h X = P H ( K ( X )) , η X i , where η X : X −→ X k , x
7→ { x } , is the K -reflection of X . Therefore, every adequate K is reflective in Top . It was shown that Sob , Top d , Top w and the Keimel and Lawson’s category are alladequate, and hence they are all reflective in Top . Some major properties of K -spaces and K -reflectionsof T spaces were investigated. In particular, it was proved that if K is adequate, then the K -reflectionpreserves finite products of T spaces. Our study also leads to a number of problems, whose answering willdeepen our understanding of the related spaces and their categorical structures.In [29], Zhao and Ho introduced a weak notion of sobriety: k -bounded sobriety. Recently, Ern´e [4]replaced joins by cuts, and introduced three kinds of non-sober spaces: cut spaces, weakly sober spaces, andquasisober spaces. In a forthcoming article we will show that some of the categories of k -bounded soberspaces, cut spaces, weakly sober spaces, and quasisober spaces are not adequate and they are really notreflective in Top .We now close our paper with the following questions about K -reflections of T spaces, where K is anadequate full subcategory of Top containing Sob . Question 6.1.
Does the K -reflection (especially, for a Keimel-Lawson category K ) preserve arbitraryproducts of T spaces? Or equivalently, does ( Q i ∈ I X i ) k = Q i ∈ I X ki (up to homeomorphism) hold for any family { X i : i ∈ I } of T spaces? Question 6.2.
Does the d -reflection preserve arbitrary products of T spaces? Question 6.3.
Does the well-filtered reflection preserve arbitrary products of T spaces? Question 6.4.
Let X = Q i ∈ I X i be the product space of a family { X i : i ∈ I } of T spaces. If each A i ⊆ X i ( i ∈ I ) is a K -set, must the product set Q i ∈ I A i be a K -set of X ? Question 6.5.
Is the product space of an arbitrary collection of K -determined spaces K -determined? Question 6.6.
Let K be a Smyth category. Is the Smyth power space P S ( X ) of a K -determined T space X again K -determined? ReferencesReferences [1] R. Engelking, General Topology, Polish Scientific Publishers, Warzawa, 1989.[2] M. Ern´e, Infinite distributive laws versus local connectedness and compactness properties. Topol. Appl. 156 (2009) 2054-2069. 156 (2009) 2054-2069.[3] M. Ern´e, The strength of prime ideal separation, sobriety, and compactness theorems. Topol. Appl. 241 (2018) 263-290.[4] M. Ern´e, Categories of Locally Hypercompact Spaces and Quasicontinuous Posets, Applied Categorical Structures. 26(2018) 823-854.[5] Y. Ershov, On d -spaces, Theor. Comput. Sci. 224 (1999) 59-72.[6] G. Gierz, K. Hofmann, K. Keimel, J. Lawson, M. Mislove, D. Scott, Continuous Lattices and Domains, Encycl. Math.Appl., vol. 93, Cambridge University Press, 2003.[7] G. Gierz, J. Lawson, Generalized continuous and hypercontinuous lattices, Rocky Mt. J. Math. 11 (1981) 271-296.[8] G. Gierz, J. Lawson, A. Stralka, Quasicontinuous posets, Houst. J. Math. 9 (1983) 191-208.[9] J. Goubault-Larrecq, Non-Hausdorff topology and Domain Theory, New Mathematical Monographs, vol. 22, CambridgeUniversity Press, 2013.[10] R. Heckmann, An upper power domain construction in terms of strongly compact sets, in: Lecture Notes in ComputerScience, vol. 598, Springer, Berlin Heidelberg New York, 1992, pp. 272-293.[11] R. Heckmann, K. Keimel, Quasicontinuous domains and the Smyth powerdomain, Electronic Notes in Theor. Comp. Sci.298 (2013) 215-232.[12] R.-E. Hoffmann, On the sobrification remainder S X − X , Pacific Journal of Math. 83 (1979) 145-156.[13] R.-E. Hoffmann, Sobrification of partially ordered sets, Semigroup Forum 17 (1979) 123-138.
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